Statistical Physics of Crystal Growth

Yukio Saito Department of Physicsl Keio University %kohama, Japan

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Library of Congress Cataloging-in-Publication Data Saito, Yukio, 1948Statistical physics of crystal growth I Yukio Saito. p. cm. Includes bibliographical references and index. ISBN 9810228341 1. Crystal growth. 2. Statistical physics. I. Title. QD921.S23 1996 548.5-dc20 96-27446 CIP

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Preface The study of crystal and its growth has a long tradition. The art of crystal growth is now well developed to manufacture various high quality crystals for electronics, but its scientific understanding is still developing. The modern technologies and observation methods of the atomic scale make possible and also require the microscopic understanding of the growth mechanisms to control the quality of thc product. Since the crystal growth is a typical example of a system far from equilibrium which involves the first-order phase transition, its comprehension involves many disciplines. In these two decades novel theoretical contributions arc given from the field of statistical physics: Kosterlitz-Thouless phase transition of the surface roughness in equilibrium surface structure, and the microscopic solvability criterion for the selection of the dendritic crystal growth. This note tries to give the concise and precise overview on the fundamental scicntific aspects of crystal growth from the simplest phenomenological argument to the latest discoveries. Technology and art of crystal growth arc omitted in this book. Important coupling of the hydrodynamics to the crystal growth arc beyond the capability of the author and arc not touched upon. The note is based on my introductory lectures on the fundamental and physicomathematical aspects of the crystal growth given at a winter school of Japanese Association of Crystal Growth and in graduate courses in many universities in Japan; Keio, Ochanomizu, Hokkaido, Kyoto and so on. I am especially grateful to H. Muller-Krumbhaar. From discussions and a longlasting collaboration with him, I learned most of thc fundamental theorics of crystal growth described in this book. The collaborations and enlightening discussions with C. Misbah, E. Brener, D. Temkin and M. Uwaha arc also deeply acknowledged. To M. Uwaha I am thankful for his reading through and giving comments on the manuscript. I am also indebted to T.Ohta, K.Wada, M.Kitilmura, who have invited me to their universities and given chances to teach thc courscs which cventually resulted in this book. Lastly I dedicate my heartful appreciation to late Prof. R. Kubo, who introduced me to the fascinating world of statistical physics. Yukio Saito

Contents

viii

16 Two-Dimensional Nucleation and Growth

52

17 Asymmetry in Attachment Kinetics 17.1 Schwoebel Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Structure of an unstable step . . . . . . . . . . . . . . . . . . . . . .

58 59 61

IV

66

Diffusion-Limited Growth: Pat tern Formation

18 Diffusion Equation

66

19 Flat Interface

68

20 Spherical Crystal

71

21 Parabolic Crystal

72

22 Stability of a Flat Interface

77

23 Fractal Dendrite 82 23.1 Diffusion-limited Aggrcgation (DLA) . . . . . . . . . . . . . . . . . . 82 23.2 Fractal-to-Compact Crossover of an Aggregate in a Finitc Gas Density 84 24 Capillary Effect and Regular Dendrite 24.1 Velocity Selection by Microscopic Solvability . . . . . . . . . . . . . . 24.2 Tip Stability and Sidebranches . . . . . . . . . . . . . . . . . . . . . . 24.3 Dendrite in a Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 Morphology Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . .

86 86 93 97 98

25 Unidirectional Solidification from the Solution 25.1 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Stability of a Flat Interface . . . . . . . . . . . . . . . . . . . . . . . 25.3 Eckhaus Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 Fully Nonlinear Behavior . . . . . . . . . . . . . . . . . . . . . . . .

101 102 105 108 111

26 Eutectic Growth 26.1 Lamellar Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Parity Breaking and Oscillation . . . . . . . . . . . . . . . . . . . . .

114 115

27 Diffusion Effect on Polyhedral Crystal: Berg Effect

120

V

Appendices

119

124

1

Introduction

We are living in the age of informatics; In computers a mass of information is handled in high speed by integrated circuits built on a semiconductor crystal. By means of a laser light emitted from semiconductor crystals, information written in a compact disc at high density is read out. Through the optical fiber runs information as the optical signal with high density and in the extremity of speed. In this information technology, semiconductor crystals with high quality are required. There are also many other crystalline materials utilized in our life; steal for cars, quartz in watch, sugar and salt and so on. To fabricate crystals under our control we have to know the mechanism of crystal growth. There are also various crystals grown in nature; minerals, gems as diamonds, snow etc. They contain information about the history how and the environment in which they are grown: Snow is called a letter from the sky, diamond a letter from deep in the earth. In order to understand their messages we have to know the dynamics of crystal growth. Crystal has an ordered arrangement of atoms or molecules in microscopic scale. The microscopic regularity shows up in the symmetry as is observed in various diffraction patterns. Also the regular arrangement of atoms brings about the symmetry in crystalline shape: Some crystals take simple forms as polyhedra as shown in Fig.1.1, reflecting their symmetry. Some crystals have complicated forms as dendrite, as shown in Fig.l.2 and in Fig.l.3. They are complex in the sense that all the snow flakes, for example, look similar with six arms but none of them are completely the same. We want to know how the crystal shapes are determined. The difference in crystal shape should be brought about by the difference in the controlling mechanisms of the growth dynamics. The relation between the growth mechanism and the resulting growth morphology is to be explored. There are many textbooks and monographs on crystal growth [74,48, 120, 851. I intend to give in this book a concise but precise overview on the fundamental theories on crystal growth from the viewpoint of statistical physics, especially on the recent developments in the pattern formation in the diffusion field. In Part I, the ideal growth formulae are derived from a thermodynamical point of view. "Ideal" here means that all the thermodynamic driving force for the phase transition is invested for the crystal growth. The growth formula thus gives the maximum growth velocity. In reality there are many hindrances against the growth. The largest effect takes place at the crystal surface, where the growth process takes place. In Part 11, the equilibrium structure of the crystal surface is studied microscopically, and the surface phase transition of roughening is discussed. For an atomically smooth surface, surface kinetics governs the crystal growth and the growth laws in this situation is discussed in Part 111. For an atomically rough surface, material transport or the heat conduction controls the crystal growth. In this case, macroscopic morphology of the crystal surface is strongly influenced, and complex pattern formation is induced. The topic will be discussed in Part

Iv.

2

1. Introduction

( a) (b) Figure 1.1: Polyhedral crystals of (a) NaCl [79] and (b) quartz. 1.Sunagawa).

(Courtesy by

(b) Figure 1.2: Dendritic crystals of (a) snow and (b) ice. (Courtesy by Y.Furukawa). (a)

Figure 1.3: Irregular dendrites of (a) MnOz (Courtesy by N.Osada) and (b) Au on P t ( l l 1 ) [41].

3

Part I

Ideal Growth Laws Crystal growth is an example of a dynamical first order phase transition. A stable phase, crystal, grows out from a metastable phase, melt or vapor. The driving force for the growth is the chemical potential difference of the stable and the metastable phases. The simple assumption of the linear response that the growth velocity is proportional to the driving force gives an ideal linear growth laws. When the crystal is finite, or when the interface deforms, the surface tension has to be included in the thermodynamics of the crystal growth. Surface tension plays an important role in the determination of the shape of a finite sized crystal, or the deformation of the flat interface.

2

Melting and Solidification: First-Order Phase Transition

At low temperatures almost all the materials order in crystalline form where atoms are arranged regularly. (Fig.2.la). This is the configuration with the minimum interaction energy E of atoms or molecules. At a high temperature atoms break the regular structure since it is less free and with little entropy. In an irregular arrangement as in liquid (Fig.2.lb) and in gas (Fig.2.lc), molecules move more freely and thus gain the entropy in spite of the energy cost. For a given temperature T and pressure p , the equilibrium state is determined by the second law of thermodynamics

0

Figure 2.1: Arrangement of atoms (a) in a crystal, (b) in a liquid and (c) in a gas phase.

Part I. Ideal Growth Laws

4

P Crystal

/

G Liquid

c

TM

T Figure 2.2: Equilibrium phase diagram in the temperature T and the pressure p phase space.

T

Figure 2.3: Isobaric variation of the Gibbs free cnergies of the liquid, GL, and of the crystal Gs as functions of temperature.

+

[121] such that the Gibbs free energy G = E - T S pV is minimized. Here V is the volume of the system. At low temperatures with small values of T the entropy S contributes less than the enthalpy H = E pV and the crystal with low H is realized (Fig.2.2). At high temperatures, on the other hand, the entropy S gives the dominant contribution and the configuration with a large S has the low free energy G: the liquid or gas phase is realized. Two phases can coexist on a coexistence curve.

+

We consider in the following a liquid-crystal phase transition as an example. Isobaric variation of the Gibbs free cnergies of liquid and crystal, GI, and Gs respectively, are shown in Fig.2.3 as functions of temperature T . At a low temperature GL lies higher than Gs showing that the crystal is stabler than the liquid, but on increasing the temperature, GL decreases faster than Gs due to the large entropy S L of liquid GI, and Gs cross with each compared to SS of crystal. At a melting point TM(~), other Gs(TM,P)= GI,(TM,P), (2.1) and the liquid becomes stabler for T > TM. On heating the crystal under a constant pressure, its temperature first increases, as shown in Fig.2.4. At a melting point TMthe temperature stops increasing and the applied heat is consumed to change the state of the matter from crystal to liquid. Since the absorbed heat does not appear explicitly as a temperature rise, it is called the latent heat L. The first law of thermodynamics says that the applied heat changes into the work pdV done to thc environment and the increment of the internal energy d E . Since the pressure is kept constant, the heat changes into the cnthalpy H = E + p V : d H = d E d(pV) = d E pdV. Thus the latent heat observed at a melting point corresponds to the enthalpy difference of the two phases as L = HI,(TM,~)-H~(TM,~ with HS and HI, being the enthalpy of the crystal and liquid phases respectively.

+

+

3. Crystal Growth from the Melt

5

Figure 2.4: Isobaric variation of the temperature due to the heating, for example, by an electric current I under the voltage V . At the melting temperature T M ,crystal starts to melt, and until the completion of melting, system absorbs the latent heat L. Since Gibbs free energies of both phases are equal at the melting point, Gs = HSTMSS= GL = HI, - TMSL, the latent heat is proportional to the entropy difference A S = SL- Ss: L = HL - Hs = TMAS. (2.2) According to thermodynamics [121], entropy S is the temperature derivative of the Gibbs free energy as S = -(dG/dT),. Thus the phase transition with a latent heat is associated with a discontinuity in the slope of the Gibbs free energy (E?) = .- L

(25)p-

P

Ehrenfest named this type of phase transition with a discontinuity in the first derivative of G as a first-order phase transition .

3

Crystal Growth from the Melt

Now let us cool the liquid to a temperature T below the melting point T M . The Gibbs free energy of a crystal Gs(T,p) is lower than that of a liquid G L ( T , ~as) shown in Fig.2.3, and the true equilibrium state is a crystalline phase. However, the whole liquid cannot instantaneously turn into the crystal. We often experience that the liquid is supercooled for a long while near the melting point. Eventually, a small crystalline nucleus is formed in the liquid, and then it grows. The crystal grows by the advancement of a crystallization front. The evolution is driven by the second law of thermodynamics so as to minimize the Gibbs free energy at a given temperature and pressure [121]. The driving force of the crystal growth is the difference of the Gibbs free energies of the liquid and of the crystal phases: AG(T,p)= G L ( T , ~ Gs(T,p). ) Since it vanishes at the melting point T M one , can expand AG up to the first order of the undercooling AT = TM- T as

Part 1. Ideal Growth Laws

6

Crystal

Liquid

Figure 3.1: Schematics of the potential surface in configuration space. Crystal phase corresponds a stable phase, liquid a metastable phase, and in between is the diffusion activation energy. In the last equality we used the relation (2.3). Under this chemical driving, the crystal grows. For a liquid molecule to be incorporated into the crystalline order, it has to change the configuration. But around the liquid molecule thcrc is a high density of othcr molecules and they hinder the free motion of the molccule. It can mainly vibrate around its average position with a frequency v, which is of the order of that of the lattice vibration. In order for a liquid molecule to change the configuration drastically, it has to overcome the energy barrier Ed of the molecular diffusion, as is shown in Fig.3.1. At a temperature T a molecule acquires the energy fluctuation E,I with a probability proportional to the Boltzmann weight exp( -Ed/kBT), where kg is the Boltzmann constant. Therefore, the crystallization rate per unit time is given as vexp(-Ed/ksT). There is, however, a counter effect, namely the melting of a crystal molecule into the liquid state. Since the Gibbs free energy per molecule, called chemical potential p = G / N , is higher in the liquid phase than in the crystal phase, p~ > ps, the rate of melting is smaller than that of crystallization by a factor cxp(-Ap/keT). Here Ap is the difference of the chemical potentials between the liquid and the crystal phascs: Ap = p l , ( T , p ) - p s ( T , p ) . By the crystallization of onc molecule, the solidification front increases its height by a molecular height a. Thus the growth rate is given as

v = avexp (-$)

[I - exp

(- %)I

.

In terms of the liquid viscosity 17 or the diffusion constant D one can dcscribe the formula as

v=kt3T [l - cxp 67ra2r]

(-31 (-3 . 1 = K [l - cxp

(3.3)

where the Einstein-Stokes rclation [56] va2exp

(-5)

=D =keT

67r17a

(3.4)

4. V a p r Growth

7

is used. Here K = k ~ T / 6 is d called ~ the kinetic coefficient. The linear growth law (3.2) or (3.3) is called the Wilson-Frenkel formula for the melt growth [196, 631, and for the small undercooling AT it is approximated as

where KT = KI/kBTTM with i = L / N being the latent heat per molecule. At small undercooling AT, the growth rate is proportional to the undercooling. The growth rate thus obtained is an ideal one and valid only when the interface is atomically rough and the whole surface is accessible for the crystallization. In the actual crystal growth the roughness of the interface controls the growth rate. Also the derivation of Eq.( 3.2) lacks the consideration of latent heat and of the associated temperature increase near the interface. The effect of heat transport is important in determining the growth rate and the morphology of the crystal, but that will be discussed later in Part 111. The Wilson-Frenkel formula says that the growth velocity V should drop drastically at low temperatures, since the liquid viscosity 17 increases exponentially. In a molecular dynamics simulation of the crystal growth of the simple molecule system [39], on the other hand, the growth rate is not limited by the mobility of the atoms in the bulk liquid: The kinetic coefficient K in Eq.(3.5) is found to be proportional to the temperature. Explanation is given such that the precursor of the crystalline order is already formed in the liquid phase near the interface, and the collective motion of the liquid facilitates the crystal growth without the activation barrier [137]. Thus, general and microscopic consideration on the kinetic coefficient seems worth to be studied.

4

Vapor Growth

Crystal can grow not only from a melt but also from a gas phase, as shown in Fig.4.1. For instance, the snow is a crystallized water from the vapor (Fig.l.Za), whereas the ice is the one grown from the liquid water (Fig.l.2b). In a semiconductor industry, vapor deposition of a thin film on the substrate is an important technology to fabricate materials with a controlled design and new functions. Molecular beam epitaxy (MBE) or atomic layer epitaxy (ALE) are among these modern technologies. For a gas phase an ideal gas is a good approximation due to its low density. At a temperature T and a pressure p , the velocity distribution of a monatomic ideal gas follows the well known Maxwell distribution 11211

where v = (wz,wy,v,) is the velocity of an atom and m is its mass. In a unit time,

f = Jm --oo dw, Jm dw,]' -W

-m

dw,n(v,(P(v)=

P

l./zG&T

Part I. Ideal Growth Laws

Figure 4.1: Vapor atoms depositing on a flat surface. number of atoms irnpingc on a unit crystal surface normal to the z-direction as shown in Fig.4.1, and try to crystallize. Here n = p / k s T is the average number density of gas atoms. Inversely, there are some atoms desorbing from the crystal surface at a finite temperature T . The desorption flux is independent of the deposition flux from a gas phase, but is a function of the temperature. If the crystal is in the gas with a saturation pressure, peq(T),the deposition rate balances with the desorption rate: The desorption flux from the crystal fdes is equal to the deposition flux feq of a gas at a saturation pressure peq(T).Assuming that an atom is cubic with a linear dimension a, the net atomic flux in an atomic area a2 contributes to the crystal growth, and the atomic height increases by a. The growth rate is thus written as (4.3) Here R is the specific volume of a single moleculc R = a3. This linear growth law is called the Hertz-Knudsen's formula (77, 1161. Since the chemical potential of an ideal gas is written as p ~ ( T , p=) p&",po) kBTln(p/po) [121], the chemical potential differcnce of the gas and the crystal is - p s ( T ) = p ~ ( T , p-) pG(T,peq(T))= k~Tln(p/p,), or written as AP = PG(T,P) p = peqeA"/LBT.Thus the Hertz-Knudsen formula is represented as

+

v = Rfe,

[exp

($)

- 11 = K [exp

(&)- 11 w Km.All

(4.4)

The growth rate V is proportional to the driving force Ap for small A p . The kinetic coefficient K is now obtained as K = Ofeq. The relation (4.4) is valid only in the ideal situation of a rough surface with the fast transport of materials in the gas phase. For a smooth and flat surface, the growth law should be modified as will be described in Part 111.

5. Solution Growth

5

9

Solution Growth

It is technically hard to crystallize a material with a high melting temperature, but by solving it into some solution crystallization becomes easier at low temperatures. For example, NaCl melts at 800°C and melt growth has to be performed at a high temperature. If it is solved into hot water, the solution becomes supersaturated by evaporation and a large crystal can be grown even at a room temperature. In the case of solution growth, there should be a solute atom in front of the interface to crystallize. The probability to find a solute atom at a certain crystallization point with a unit volume a3 is u3c for a solution with a concentration c. This atom is oscillating around the average position with a frequency u, and tries crystallization. In the solution, however, the solvent molecule makes some chemical bonding with the solute molecules. Thus, for the solute molecules to be incorporated into the crystalline structure, the solvent molecules have to be desolved from the solute. For this desolvation process there is an energy barrier &-I. Among v trials of solidification, the rate which overcomes the desolvation energy barrier is given by the Boltzmann weight exp( -Edeaol/kBT). Then the velocity of crystallization is given by Vcry(c)= uv(cu3) exp( -Edesol/kBT). There is an inverse process of melting of crystal molecules. Its rate is determined by the temperature, and should balances the crystdlization from the solution with the equilibrium concentration ces: Vmel= Kry(cq). The net rate of crystallization is

The chemical potential of a dilute solution with a concentration c is expressed as keTln(c/co). The chemical potential of the crystal is equal to that of a solution with an equilibrium concentration c, as ps(T) = p.o~(T,ceq). Then the excesS chemical potential of the supersaturated solution is written as pWt(T,c)= ps0i(T,co)

+

C

A p = psol(T,c) - p s ( T ) = kBT In - M k B T Ceq

The growth rate is expressed in terms of the chemical potential difference Ap as (5.3)

The kinetic coefficient is now defined as K = vu4ce, exp(-Ede,l/kBT). The linear growth law (5.2) is valid only in an ideal case when the interface is rough and the whole interface contributes to the growth. Actually by the solidification the number of solute molecules in front of the interface decreases, and to compensate this solute deficiency, molecules have to be transported from the far end of the solution. By including this diffusional material transport, we reach to the different growth law. But the detail will be discussed later in Part VI.

Part I. Ideal Growth Laws

10

Exercise: If the transport of materials in thc ambient phase is important, it introduces a length scale 1 which characterizes thc cffectivc range for the material to be incorporate from the environment. Find in this case thc growth formula of an [100]-faceof a crystal with a unit cell a1 x a2 x u 3 . Answer: The number of atoms to be incorporatcd on a unit of [100]-face with an area ~ 2 is ~given 3 by c a a a 3 l with a solute concentration c. Sincc the interface advances a height al by the crystallization of one atom, the growth rate is described BS

v,= IvRe-'Sd=~~@"T(Cwhere R = U

6

~ U ~ C is Jthe ~

C,>J,

(5.4)

volume of the unit cell [23].

Equilibrium Shape

So far we considered the growth of a crystal with a flat and infinitely large interface. If the interfacc deforms, the intcrfacc area changes and the associatcd encrgy change has to be considered. Also in thc carly stage of crystal growth, it starts from a small embryo. During growth embryo size increases, and the energy cost at thc interface changes. Thus the interface energy has effects both on the crystal shapc and the growth dynamics. In this section we focus on its effect on the crystal shape in equilibrium [200, 121, 61. Even though the realization of equilibrium shape is difficult due to its long relaxation time, there are some expcriments with very small crystals [78,79, 1361,or with a crystal with fast material transport [147]. Due to the energy cost at the interface, a crystal nucleus with a finite size and shape cannot coexist with a mother phase at a bulk transition point, where the chemical potentials of two phascs are equal: Ap = 0. Finite A p is necessary to compensate the energy incrcase at the interface and to keep the small nucleus in the stationary state. Let us assume that z = C(x,g) describes the phase boundary between the crystal

X

Figure 6.1: Crystal surfxc.

6. Equilibrium Shape

11

and liquid phases (Fig.6.1). Then the free energy gain by the solidification of Ns atoms is given BS

where 0 and V are the atomic volume and total volume of the crystal cluster, respectively, and the volume integration is taken over the crystal phase ( z < ((x, y)). The bulk free energy Gb is negative for supercooled melt (ps < p ~ ) .There is an energy increase due to the formation of an interface. The interface free ener er area y(p,,p,) is a function of orientation of the interface n = ( - p z , -py, 1)/ 1 p; +pE where pE a
+ F

G8

= JdA7(Pz,Py) = JdzdYf(PzlPy).

(6.2)

Since the projection of the surface area d A in z-direction is equal to the area dxdy in sy-plane as n,dA = dxdy, the free energy density f per unit area in zy-plane in Eq.(6.2) is given as f(PZ,PY) = " ( P m P Y ) J l m . (6.3) The equilibrium state is determined by the stationarity condition of the total free energy G = Gb G.. The change of the bulk free energy Gb by the small variation of the interface height 6C(z, y) is written as

+

6Gb

--A'n /6<(z,

where we introduce a parameter energy varies as

y)dxdy E -2X

J -S
(6.4)

= Ap/2R. On the other hand, the surface free

where partial integration is performed to derive the last equality. By imposing the stationary condition -SGb 6G, = 0 (6.6)

+

for arbitrary variation 6<(x,Y ) ~one gets the Euler-Lagrange equation

a af 2 x + -axcap,

The solution of this equation is

a af + -ayap,

= 0.

Part I. Ideal Growth Laws

12

It shows that variables x and p , are conjugate as well as y and py are, and f(p,,p,) behaves as a potential function. The Legendre transformation introduces the new

which is a function of x and y variables as dg = df

+ d(Axpx)+ d(Ayp,)

(2

= x -ds

:)

+ -dg

= Ad<.

(6.10)

On integration one gets g(-Ax, -Ay) = A<. The potential g is thus equal to the shape

c=

( = -gA= - Af+

XPX

+ VP,.

(6.11)

By rearrangement, it reduces to (6.12) Since the normal vector to the interface is given by n = [I +p: the point r = ( 2 ,y, <) on the interface satisfies the relation

+ pi]-1’2(-pz, -A, 1), (6.13)

The analogous formula holds for a one-dimensional interface of a two-dimensional

Figure 6.2: Wulff plot and equilibrium shape.

6. Equilibrium Shape

13

Figure 6.3: Local coordinate system; z axis in normal direction, and local principal directions.

51

and

22

in the

system. Actual realization of a one-dimensional interface is a step on a crystal surface with the step free energy P(n). The equilibrium shape is determined by the relation 02P (r n) = -, AiJ +

(6.14)

where 02 = a2 is an atomic area. Since the 1.h.s. of Eq.(6.13), h E (r+n),is the vertical length from the crystal center to the interface with orientation n, Eq.(6.13) shows that the length h is proportional to the surface tension $n). By using this relation one can draw the equilibrium shape of a crystal as shown in Fig.6.2 (Wulff plot) [200]. Draw a line from a center 0 to the direction n with a radial length proportional to the surface tension r(n), and then draw a plane perpendicular to it through the end point: The crystal surface can be a part of this plane. By varying the orientation n, the envelope of the perpendicular planes gives the equilibrium shape. Another way to represent the equilibrium shape is its relation with the interface stiffness. In three dimensions, we choose the local coordinate ( X I , x2,z ) such that z axis is in the normal direction of the interface, and 21, 22 axes in the local principal directions with principal radii of curvatures, R1 and R2 (Fig.6.3). Then the profile of a crystal near this point is approximately given as (6.15) with X I = R141, x2 = R&. Here the angles 41 and 42 are the deviation of the normal direction as shown in Fig.6.3. Then the slopes are p ; = d z / d x ; = -x,/Ri = -8; with

Part 1. Ideal Growth Laws

14

i = 1 and 2. Equation (6.7) can be expressed as (6.16) Herc

f =7

( 8 1 , 8 2 ) 4 m ,and thc surfacc stiffness at

51

= x2 =

= 82 = 0 is

(6.17) The curvature is (6.18) Thus Eq.(6.16) reduccs to the relation (6.19) If the surface tcnsion is isotropic, then with the critical radius

;U;j

= y&j and thc crystal nuclcus is spherical

270 R, = -, (6.20) AP obtained from Eq.(6.19) with R , = R2 = R,. This radius corresponds to the critical radius of a three-dimensional nucleus in equilibrium with thc undercooling or supersaturation, A p .

Exercise 1: In two dimcnsions the interface is defined by the arc lcngth s and the angle 8 of thc normal vector n from ?/ axis. (Fig.6.4) (1) From Eq.(6.14), show that thc point

( 2 , ~on )

the interface satisfies the relation

k ( s ) = p(8)sind +@'(8)cosO

(6.21)

Xg(s) = P ( 8 )cos 8 - p'(8) sin 8

(6.22)

with p'(9) = dp/d8 and X = Ap/Rz.

Figure 6.4: Geometry of a stcp at a sitc r with a normal vector n. s is the arc length and 8 is the angle of thc normal vcctor.

15

6. Equilibrium Shape

(2) Find the relation A=-

P

(6.23)

P

(6.24) and the radius of curvature p = ds/dO. Answer:

(1) Since n = (sin0,cosO) from Fig.6.4, Eq.(6.14) can be written as

B

(r n) = z s i n d + ycosd = -.

X

(6.25)

By differentiating both sides by s, one obtains

dx

-sind+dycosd+(xcosd-ZIsinQ)ds

ds

df3

p’dd ds = X ds

(6.26)

From the geometry shown in Fig.6.4, dx = dscosd,

d y = -dssinO,

(6.27)

and one gets zcosd - ysind = -.pl

(6.28)

x

Solving (6.25) and (6.28), the desired relations (6.21) and (6.22) are obtained.

(2) Differentiating Eq.(6.28) with s, one gets d?i . -ccosd--sind-(ssind+ycos~)dS ds

dx

dQ P”d6 = ds X ds

(6.29)

Inserting the values of dxfds and dylds from Eq.(6.27) and by using the relation (6.25), one gets the relation

P

dd ds X

-- =cos28+sin2d= 1.

(6.30)

By rearrangement, the desired result (6.23) is obtained.

Exercise 2: Draw the equilibrium shape of a two-dimensional crystal with a tension

Part I. Ideal Growth Laws

16

Figure 6.5: Polar plot of the step free energy P = Po(l+ Zcos48) in a dashed curve, and the equilibrium shape in continuous curve. (a) E = 0.06, and (b) E = -0.06. Answer: From Eqs.(6.21-6.22), the shape is determined by the parametric representation;

x

-x

= (1 +Fcos48)sinO -4Tsin4OcosB

PO

x

--y

= (l+EIco~48)~0~8+4Fsin48sin8.

(6.32)

PO

The shape is plotted in Fig.6.5. For C > 0, the direction 8 = 0 corresponds to the maximum of the surface free energy P and to the minimum of the stiffness 8. The equilibrium shape has a pointed corner there because the curvature is large.

Exercise 3: Often a lattice model is used for a simple description of a crystal shape. The space is divided into a regular lattice and each lattice site can be empty or be occupied by a crystal atom. For an occupied site an king spin variable Si = 1 is associated, and for an empty site S, = -1. There is cohesion between the neighboring crystal atoms. If the bond from a crystal atom is not connected to the other atom, this broken bond causes an energy cost J . This energetics can be represented by the Ising ferromagnetic Hamiltonian [121]: J NzJ 7-l= -sjsj (6.33) 2 4 .

c

+

Here z is the coordination number, and z = 4 for a square lattice model. There the critical point is exactly known to he at I1491 (6.34)

7. Growth Shape

17

Figure 6.6: Equilibrium shape of 2D Ising model at various temperatures below T,. Normalized temperatures are T/T,=O.l (outermost) to 0.9 (innermost) with temperature differences 0.1. The exact form of the interface free energy P ( 0 ) is also known at a temperature T as a function of the orientation angle 0 from the (01) direction as [159]

'(') -

ksT

I

- cosOlsinh-'(al cosO1)

+ I sinBlsinh-'(al

sine[)

(6.35)

with 1 - b2

1

+ h i n 2 20 + b2 cos220

(6.36)

and (6.37) Draw the equilibrium shape of a square Ising crystal at various temperatures by using this interface energy [159,71. Answer: Using the parametric representation, (6.21) and (6.22), the profile is obtained straightforwardly as shown in Fig.6.6. On increasing the temperature the shape becomes isotropic and the size becomes small.

7 Growth Shape According to the general formalism of nonequilibrium thermodynamics, velocity of the time evolution of a dissipative system is proportional to the thermodynamic driving force given by the gradient of the free energy [142]. Thus the time evolution of the interface z = C(z, v,t) is written by

18

Part I. Ideal Growth Laws

Figure 7.1: Growth shape by Wulff plot using kinetic coefficient K. In equilibrium Eq.(7.1) reduces to the stationarity condition (6.6) for the equilibrium shape. By choosing the local coordinate such that the t axis is oriented in the interface normal n, one obtains the evolution from Eqs.(6.4-6.5) and (6.16-6.18) as

where we show explicitly the possibility that the kinetic coefficient K depends on the orientation n. The growth law (7.2) is similar to the ideal growth formulac, (3,5),(4.4) and (5.3), but thc chcmical driving force Ap is modified by the curvature effect. At the pointed part with positive curvaturcs, R , > 0 and Rz > 0, the growth rate decreases by the surface tcnsion effect. This is called the Gibbs-Thomson effect. For a sufficiently large crystal thc radius R1 and R, arc very large, and the last term related to thc interface stiffness may be neglected. Witliout it and assuming that Ap is constant d1 over the interface, onc can easily integrate Eq.(7.2) up to the time t as (7.3) where r is a point on thc interface orientcd in n direction. Eq.(7.3) is similar to the cquilibrium shape Eq.(6.13), with the interface tension r(n) being replaced by the kinetic coefficient K(n) [46]. Therefore, wc can draw the growth shape of a crystal by the Wulff plot of K(n) instead of the r(n),as shown in Fig.7.1. Also one can derive the reIation between thc local curvature and the kinetic stiffness I?;.j = KS;j+82K/a&8@,. In an isotropic case, thc crystal grows spherically with its radius given by (7.4)

7. Growth Shape

19

If the orientation n-dependence of the kinetic coefficient K is so strong such that a corner appears in the kinetic Wulff plot, the Gibbs-Thomson term with surface tension is no more negligible. The growth shape with both surface tension and kinetics are generally treated by Miiller-Krumbhaar et al. [142] and by Uwaha [184].

Part I1

Statistical Mechanics of Surface So far the ideal growth formulae of the crystal are discussed in a macroscopic, thermodynamic sense. In this and the next parts, we consider the microscopic aspects of crystal growth. Since the growth takes place at the crystal surface, it is quite natural to imagine that the surface structure influences the growth. F’rom a microscopic model of the surface configuration, a surface phase transition, a roughening transition 143, 194, 1951, is found: At low temperatures surface is flat and smooth, whereas at high temperatures it is rough in the atomic scale. The roughening phase transition induces the faceting transition in the equilibrium crystal shape.

8

Position of Crystallization: Kink Site

Before considering the crystallization in an atomic level, we have to identify when or in what situations an atom is said to be crystallized. In the gas phase an atom is free and makes no bond with other atoms. In a crystal it makes z bonds with neighboring atoms to lower the energy. An energy gain for each connected bond is set -25. For N crystallized atoms, there are totally 2N/2 bonds, and the total cohesive energy is - J z N , or -22 per each crystal atom. After melting all the atoms become free without any bond connections, and the energy is zero. By assuming that the work associated to the volume change is negligible, the enthalpy variation by the melting or the latent heat per atom is given by 1 = 2.7. If an atom freely moving in the gas phase impinges on the crystal substrate and makes a bond of 2 / 2 nearest neighbors with the crystal substrate, there is an energy gain of z J , and the atom can be regarded to be crystallized. The problem then is when an atom will acquire 2 / 2 bonds. On a completely flat crystal surface, the adsorbed atom cannot make so many bonds. But on a crystal surface, there are various defects as shown in Fig.8.1. A flat portion is called a terrace. When the heights of two consecutive terraces differ, there is a step between them. A step can be straight as it runs in a closed packed direction. It can also bent and change the orientation at a kink position. When nn atom impinges on a flat terrace, the atom makes only a few bonds with the underlying atoms. The bond connection is so weak that the atom can meander on the surface. This atom is called the adsorbed atom, or adatom for short, and performs surface diffusion. During meandering, the adatom may come in contact with an uprising step. Then step provides some additional bonds parallel to the terrace, but the number of bonds is still in short of 2/2. The adatom can slide along the step edge by edge diffusion, and may reach to the kink position where there are 2/2 nearest neighbors. At this kink site, the atom is really said to be crystallized. The striking feature of the kink site is that it never disappears by crystallization or melting; it only slides along the step. For the fast crystallization, therefore, it is

8. Position of Crystallization Kink Site

21

/w - // vacancv

LU

n

terrace stepoverhang

a

Cluster

I/

kink

Figure 8.1: Surface configuration with terraces bounded by a step with kinks. desirable to have many kink sites or steps on the surface. But the surface with many steps and kinks is rough with many broken bonds, and it has high surface energy. The surface cannot be rough at low temperatures where only the minimum energy configuration is allowed. The rough surface is possible only at high temperatures. The roughness of the surface is expected t o change drastically at a certain temperature. Since the surface roughness controls the growth mode of the crystal, the roughening phase transition of the surface will be discussed in the next section.

Exercise: In a lattice gas model for crystal growth, a density variable ni at a lattice site a takes a value 1 when the site is occupied by an atom, or it is 0 when the site is empty. If a broken chemical bond from a crystal atom costs an energy J , the interaction energy is expressed as 8=J

[ni(l - nj)

+ (1- ni)nj]-



where < zj > means the summation over the nearest neighbor sites pairs, and Ap is the chemical potential gain by crystallization. Show that the Hamiltonian (8.1) can be transformed to the two-dimensional Ising Hamiltonian

Here an Ising spin variable Si = 2n; - 1 takes values +1 or -1, and the field is H = Ap/2 [128].

Part II. Statistical Mechanics of Surface

22

(4

(h)

Figure 9.1: Surface configuration with steps observed by scanning tunneling microscope (STM). (a) Si(ll1) [192]and (b) Si(OO1) faces [178]. (Courtesy of M.G. Lagally).

9

Surface Roughening

We study the equilibrium configuration of the crystal surface in contact with the ambient gas phase, which is reali.md on the vapor-crystal coexistence curve in equilibrium phase diagram (Fig.2.2). Along this curve the chemical potentials are the same for crystal and gas phases. The temperature variation affects only the surface free energy associated to the surface configuration or its roughness. At low temperatures the crystal is polyhedral, as shown in Fig.l.1, enclosed by atomically flat faces with low Miller indices, called facets, since these faces have low energies. But the flat face has only a single possible configuration, and has no entropy. If atoms jump out from the flat face and sticks on it, there is an entropy gain S connected to the possible numbers of positions for atoms to be taken out and to be put on. But there is an energy cost E , and the equilibrium configuration at a finite temperature T is the one which minimizes the free cnergy F = E - T S . Thus the surface structure is determined by compromising two contributions, E and S [194, 195, 1461.

9.1

Monte Carlo Simulation

In order to understand the atomic structure of the crystal surface, it is good to see the surface configuration: Seeing i s believing. Recent development of various microscopes as scanning tunneling microscope (STM) and others make it possible to see the surface structure in an atomic level, as shown in Fig.9.1. Theoretically, it is also possible to produce atomic configurations of the surface by using various computer simulation methods. One method is the Monte Carlo simulation [25, 761, which produces various microscopic configurations stochastically under the influence of thermal fluctuation. Equilibrium configurations of a surface are obtained by this method as shown in Fig.9.2 [194]. By way of solid-on-solid (SOS) model for the crystal growth (Appendix A9.1),

9. Surface Roughening

23

Figure 9.2: Surface configurations by Monte Carlo simulation; Parameters are the temperatures LBTIZJ. [194] Monte Carlo simulation procedure is now explained [69]. Crystal surface configuration is assumed to be described by the integer height variables { h } G {h(l), . . , h( N ) } on a square lattice with N = L2 lattice sites. With the thermal fluctuation, the configuration { h } is realized stochastically with a probability P ( {h } ,t ) at time t. The height h(i) at site i changes to another height h’(i) with a transition probability w(h(i) + h’(i)) per time. Then the time variation of the probability P in a small time increment At is given by

-

P (h(1). * * h(i)* .. h ( N ) ,t + At) = P (h(1 ) .. . h(i)* . . h ( N ) ,t ) N

- C ~ ( h ( i-+) h’(i))AtP(h(l)...h(i)...h(N),t) i=l

N

+ C w (h‘(i)+ h ( i ) )A t P (h(1). - . h‘(i)

* * *

h ( N ) ,t ) .

(9.1)

i=l

The second term represents the probability decreasc due to the probability escape from { h } to other configurations, and the third term represents thc increase due to the probability income from other configurations to { h } . In the continuous time limit,

24

At

Part 11. Statistical Mechanics of Surface + 0,

one obtains the master equation of the probability; N

N

a p ( { h } , t )= - C w ( h ( i ) + h’(2)jP({h},t)+ C W ( l l ’ ( i ) + h(Zj)P({h}i,t).

dt

i=l

r=l

(9.2) Here { h } *is a configuration different from { h } only by a height at the site i. In order to realize thermal equilibrium asymptotically (t -+ co),the Boltzmann distribution (9.3) should be a stationary solution of the master equation. Here X ( { h } )is the Hamiltonian defined as

C [m(i)-h(j)l+A~,h(i)=

‘H({h})=J

E({h})+ApCh(ij,

(9.4)

i

i



with J being the energy cost of a broken bond, A p = p~ - ps the chemical potential difference between the gas and the crystal phases. If the transition probability w satisfies the detailed balance condition;

the equilibrium is confirmed. The crystallization ratc at the site .i is chosen to depend only on the chemical potential difference as

w(h(i)

--$

h(i) + 1) = exp

(3 -

.

The evaporation rate at the site i is determined from the detailrd hlancc condition (9.5) as

w(h(2)

-.+

h ( i ) - 1) =

AE

Here A E = E(h(z) - 1) - E(h(s))is thc variation of the hond cncrgy associatcd to the lowering of the height h ( z ) , and is given by A E = -25(7t - z/2) with n being the number of nearest neighbor sites with hciglits lower than h ( z ) and z is the coordination number. Surface configurations with corresponding n are shown in Fig.9.3 for a one-dimensional cxamplc ( z = 2). Since the maximum of n is z, the T h m by choosing maximum evaporation ratc is proportional to &IkHT.

the transition rate uiAt in Eq.(9.1) ncvcr exceeds unity.

9. Surface Roughening

25

n=O

I

AE=2J

0

2 -25

Figure 9.3: Configurations of a one-dimensional interface ( z = 2) before evaporation of the shaded top atom on the i-th column. n means the number of nearest neighbor sites with the height lower than hi, and AE is the variation of the bond energy. The real configuration change of the simulation is performed as follows: Select a lattice site i randomly, and calculate n and various transition probabilities wAt. Then, create a random number u which is uniformly distributed between 0 and 1. If u lies between 0 and AteAPlkWT,the height h(i) is increased by 1. If u lies between AteAPlkBTand At(eAPlkBT +e-AElkwT),the height h(i)is decreased by 1. Otherwise, the height does not change. After trying the height variation N times, every site has been tried once on average to change its height, and the one Monte Carlo step (MCS) is said to have been performed. The time increases by At. See the sample program in Appendix A9.2. The average height at the m-th MCS is defined as

where h(i;m)represents the height at a site i at m-th MCS. In equilibrium where Ap = 0, the height should remain constant, and the thermal average of height is obtained by 7

( h )=

c

Mn+M

(h(m)).

(9.10)

m=Mo

Other quantities are calculated similarly. In some cases, the Monte Carlo procedure can be regarded not merely a mathematical algorithm to obtain equilibrium quantities, but also as a physical process of relaxation dynamics. For example, the SOS simulation so far explained can be regarded as to mimic a crystal growth from a homogeneous ambient gas phase. In order to incorporate the kinetic coefficient K , the time increases At/K after one MCS. The growth rate (5.2) with additional curvature correction is obtained from the simulation by K (9.11) V ( t )= ( ( h ( m+ 1))- ( h ( m ) )1)

at

where the time is t = mAt/K.

Part II. Statistical Mechanics of Surface

26

At very low temperatures ( k s T << J) with a small supersaturation, the time increment At defined in Eq.(9.8) is so small that configuration change occurs very seldom. In order to circumvent the difficulty and alter the configuration efficiently, the waiting time method is often used. Details are found in references (69, 188, 881. Simulation results of the equilibrium configuration are shown in Fig.9.2 [70]. At low temperatures, the surface is Aat with few bumps and holes, whereas at high temperatures the surface is rough. Thus the change of the surface configuration by temperature is obvious. This surface structure change is in fact a phase transition, which takes place at a critical temperature. For the real understanding of the phase transition, one needs analytical theory on it.

9.2

Mean Field Theory by Jackson; a-Parameter

For an analytical study on the roughening transition, we start from the simplest model, where the interface is assumed to consist of a single layer between the semiinfinite crystal and the vacuum phases [43, 901. The interface layer is divided into a two dimensional lattice cells, and each cell site can bc occupied by a crystal atom or can be empty. Between the neighboring crystal atoms a cohesive energy of attraction is assumed. If the layer is almost empty, the layer belongs to the vacuum phase and the crystal phase terminates sharply before the interface layer: The crystal has a sharp interface. If the interface layer is almost fully occupied, the layer belongs to the crystal, and the crystal terminates sharply after the interfacc layer. Only when the half of the layer is empty and the other half is occupied by crystal atoms, the interface layer belongs neither to the crystal nor to the vacuum phase. In this case the interface can be regarded to be rough. Burton, Cabrera and Frank treated the square lattice case exactly [43], because the model is equivalent to thc exactly soluble king ferromagnet as (8.1) or (8.2). Jackson later treated the same model in the mean field approximation [go], and estimated the roughening temperature TR , and compared it with the melting temperature T M .Here I summarize briefly the latter mean-field treatment. The interface layer consists on N cell sites, and each cell site has z8 nearest neighbor sites in the layer. When there are NS crystal atoms on the interface layer, the energy increase is estimated as follows: Among 2, neighbors of each crystal atom, (1 - Ns/N)zq of them are expected to be empty in the mean field approximation. Therefore, the total energy cost E is proportional to the total number of broken crystal bonds as E = Jz,(l- Ns/N)Ns. (9.12) The entropy S is obtained as the logarithm of the number W of the microscopically different configurations. Since W is the number of ways to choose Ns sites from N lattice points to locatc crystal atoms, it is easily obtained as W = N!/[Ns!(N By using the Stirling’s formula In N ! M N In N - N , the entropy is obtained as

.I!)&-

S = k g l n W M ks[NIn N

- NslnNs - ( N - Ns)ln(N - Ns)].

(9.13)

9. Surface Roughening

27

0

F

0

0.2

0.4 0.6 0.8 Y‘= N s / N

1

Figure 9.4: Free energy F as a function of the parameter @ = N s / N at various temperatures. Below the roughening temperature TR,F has double minima, whereas above TRit has a single minimum. The dependence of the free energy F = E - T S on the parameter 3 = N s / N is shown in Fig.9.4 at various temperatures. By minimizing the free energy F , we obtain the equilibrium value of the parameter 9, or the degree of crystallinity of the interface. At a temperature below the roughening temperature (9.14) there are two minimum points in F - @ diagram, showing that the interface is almost empty (6 << 1) or is almost full (Q M 1). At high temperatures there is a single minimum at the intermediate value Q = 0.5; The interface is rough. Since the coupling energy J is related to the latent heat per molecule by 1 = zJ with L being the bulk coordination number, the roughening temperature can be exDre5sed (9.15) The comparison of the melting temperature TMand the roughening one TRyields the following parameter LE 2TR -(9.16) ZkBTM TM ’ which is readily evaluated by using experimental data. If 01 < 2 the melting point TM is higher than the roughening temperature TR and the interface at TMshould be rough. If Q! > 2, on the other hand, TMis below the roughening temperature and the interface is flat. The parameter o is called the Jackson’s o parameter [go], and is used t o classify the material for surface roughening or to guide experiments.

(-

)

28

Part 11. Statistical Mechanics of Surface

This model is equivalent to two dimensional king ferromagnet defined in (6.31), and the exact solution is known (1491.The singularities in thermodynamic quantities are well studied, and for example, the specific heat shows a logarithmic divergence. But the single layer model of interface cannot correctly describe the roughening transition. In this model the height difference at two separated points is 0 or 1 by definition, and the interface is always flat. If the interface is really rough, heights at two distant points should bc uncorrelated and fluctuate strongly to diverge at infinite separation. Burton, Cabrera and Frank extended the model with multiple layers and analyzed it in the Bethe approximation [43].They found no singularity in the free energy, meaning the absence of phase transition when the height fluctuation increases. The Bethe approximation is better than a simple mean field approximation, but as for critical phenomena the same qualitative behavior is expected since both approximations neglect the effect of fluctuation. For the analysis of the true roughening transition, one should include the possibility of arbitrary height difference at two separated points. The thermal fluctuation influences strongly on the phase transition, and one needs sophisticated method as variational [I631 or renormalization group method 1114,49, 1461. The critical behavior of the phase transition turns out to be quite different from that of the single-layer Ising model. However, the transition temperature is quantitatively not much different from the two-dimensional Ising model, and thus the Jackson's &-parameter is used as a convenient criterion for surface roughening.

9.3

M e a n Field Argument in terms of Steps

We now look the surface roughening from another point of view, in terms of steps on a crystal surface. For a rough surface there are two- dimensional nuclei, where the inside terrace is one atomic height higher or lower than the exterior one, as shown in Fig.9.5. For a nucleus with the perimeter step length L , the energy cost is E = J L / a due to the broken bonds. Here a is the linear size of the atom. As for the entropy, one counts the possibility of step configurations with a fixed perimeter length L. A stcp looks quite similar to a polymer with L / a segments. Each segment should be oriented in onc of the z, directions of nearest neighbor connection. The first segment can be put in one of z, possible orientations. From the second segment on, it can be in one of z, - 1 orientations, because it cannot fold back on the previous segment. Then the entropy is approximately given by S M ksln(z, l)L/" = ks(L/a)ln(z, - 1). Herc we have neglected such restrictions that the step loop should be closed and cannot cross itself. The free energy is then obtained as F = [ J - kBTln(z, - l)]( L / a ) = PL, with P being a step free energy per length. At a low temperature below the roughening temperature TR = J/k* ln(z, - l), p is positive and the state which minimizes the free energy F contains no step ( L = 0): The surface is flat. At a high temperature (T > TR)L = 00 corresponds to the state of the minimum free energy and the surface becomes rough with an infinitely long step running. In this case the step-step interaction becomes important and the step

9. Surface

29

Figure 9.5: (a) A two-dimensional nucleus bounded by a step with length L. (b) Top view of a step loop. free energy is shown to remain zero.

9.4

Short Summaries of Roughening Transition

In order to understand the roughening phase transition correctly, the mean field approximation is insufficient and the fluctuation effect has to be considered carefully. Treatments by the variational method [163] and by the renormalization group theory [114, 49, 145, 1461 are described in Appendices A9.3-4 in detail, and here we give a short summary of the renormalization results. There are also some exactly soluble models of the surface structure [187, 931 which shows the roughening transition, as is described in Appendix A9.5. The result is also summarized here. In the SOS model the interfacial configuration is described by the height variable hfr) at a two-dimensional site r. Below the roughening temperature TRthe interface is flat and the height fluctuation is small such that the height difference correlation

-+ rd - h(ro)I2)

G(r) =

(9.17)

at two separate points remains finite. On the other hand, above TR the interface is rough and the height fluctuation is so large that G ( T )diverges as T --t 03. According to the renormalization group calculation [146], G ( T )diverges logarithmically above TR. Below TRthe height fluctuate logarithmically at short distances within the correlation length (, but for large distances T > the fluctuation remains constant:

<

G(T)

{

for T < 5 for T >

<

(9.18)

Part 11. Statistical Mechanics of Surface

30

The corrclation length diverges at the roughening point TR as (9.19) Within the scale of correlation length [, the surface is rough, as Eq.(9.18) shows. This means that steps with a perimeter length up to can be excited by thermal fluctuation. By denoting the step free energy per length as 0,the total step free energy p( of a step with a length should be of the order of the thermal energy k s T . Therefore, the step free energy p is given by

<

<

CI

(9.20)

The singularity of the step free energy is an essential one, whose derivatives of arbitrary order vanish at the transition point TRand there is no divergence in derivatives, the correlation length is infinite and thus p = 0. such as in specific heat. Above TR, There are some special models of a crystal surface, which arc exactly soluble [187, 931. Thcy arc mapped onto the inverted F-model, a special case of a six-vertex model [129]. The step free energy is shown to have the same singularity with that predicted by the renormalixation group method, Eq.(9.20). Thc Monte Carlo simulations also show that the height difference correlation funcand it saturates below TR [127, 1641. tion G ( r )diverges logarithmically above TR,

<

10

Step Fluctuation in Equilibrium

Above the roughening tcmpcraturc the crystal surface is rough and thcrc arc kinks cvcrywhcrc on the surface. In this case the growth formula of Hertz and Knudscn may be valid, if the transport is sufficiently fast. Below TR thc surfacc is flat and there is no thermally excited kinks. Then, how can the crystal grow? When an atom impinges on a flat surface from the ambient gas phase as shown in Fig.8.1, the adatom moves around the surfwe before it evaporates back into the gas. If an adatom mccts othcr adatoms during surface diffusion, thcy make bonds and form a cluster to gain energy. The cluster is stabler than the isolated adatom. Further coalescence of adatonis niakcs the cluster larger to forin a nuclcus as shown in Fig.9.5. Around this two-dimensional cluster or crystal nuclcus there is a step and kink positions. Thus the two-dimensional nuclcus provides the kink sites for the crystal growth. This mcchanism of growth is called the two-dimensional nucleation and growth. Thcrc is another rricchanism of crystal growth. The ~rioleculararrangcrncnt of the crystal is not always pcrfcct, but sonictimcs there arc defects. Among thcm is the screw dislocation, a line defect where the Inolccular arrangement is shifted in two consecutive lattice planes. When thc screw dislocation ends up at the crystal surface, it produces a step running from the dislocation line (Fig.lO.1). If there are many

10. Step Fluctuation in Equilibrium

31

Figure 10.1: Spiral growth by a screw dislocation.

Figure 10.2: Spiral on SiC.(Courtesy of LSunagawa.) kinks on a step, the meandering adatoms meet the step, are incorporated in the kink sites, and the step advances. But since one end of the step is pinned by the screw dislocation, the step winds up around the dislocation in a spiral form. This spiral step never disappears but grows to supply the kink position. This mode of growth is called the spiral growth (Fig.lO.1 and 10.2). The nucleation and spiral growths are two main mechanisms for the growth of the crystal with flat faces. Growth laws of the two modes, nucleation and spiral growths, will be described later. First we calculate the kink density on the step. Since the kinks are related to the change of orientation of the step, it is related to the step fluctuation, too. Nowadays step fluctuation can be observed by various atomic-level microscopes as scanning tunneling microscope (STM) [178, 192, 86, 411, atomic force microscope (AFM), reflection electron microscope (REM) [4, 125, 1261 and so on. STM and REM observations are shown in Fig.9.1 and Fig.10.3, respectively. From these observations one can derive the microscopic information of step stiffness and energy parameters. Let us consider a simple model of the step running, on average, in z direction, as shown in Fig.lO.4. By denoting a lattice constant a, the step position is written as y(i) at 2 = ia. When the step runs in the next position z = (i l)a, its height v(i + 1) can be the same with the previous one, or can be up or down by a single unit and form a kink. The total energy of a step running from z = 0 to z = L is written

+

Part 11. Statistical Mechanics of Surface

32

(4

(b)

Figure 10.3: (a) A step train and (b) an isolated step observed by reflection electron microscope (REM) [4].

as

x=J

L , - + J"la~ I~y ( i ) - y ( i - 1)1 a

i=l

a

(10.1)

I

with J,, J , being the energy cost of a broken bond in x and y directions respectively. Since the height difference (y(z) - y(z - 1)) / a between neighboring positions takes only three values, 0 or f l , one can easily calculate the partition function at a temperature T as

Therefore, the step free energy per length is obtained as PO

=

kRT1nZ = (Jy - lcBTln [ 1 + 2exp (-&)]) . 7 a

0

(i-lja ia

L

(10.3)

x

Figure 10.4: Solid-on-solid model of a step running on average in 2-direction.

10. Step Fluctuation in Equilibrium

33

The density of a kink is also obtained as (10.4)

J, thus corresponds to the kink energy. At a finite temperature, there is a finite density (Inl) of kinks on a step, and thus the adsorbed atoms reached to the step can be easily crystallized at kink sites. If J , = Jy = J , the step free energy density PO vanishes at a temperature TR = J / ( k ~ l n 2 ) .This is said to be the roughening transition temperature of this model. The correlation function of the step height difference at a separation x defined as

characterizes the roughness of the step. Since the height difference an; at two neighboring sites i and i - 1 has no correlation with those at other positions, the step fluctuation is easily calculated as xla G ( x ) = a2 c ( n 1 )= a x ( n 2 ) .

(10.6)

i=l

The height correlation diverges in proportional to the separation x as x increases, and the step is rough at any finite temperatures. In the present model, (n2)= (In[) and the coefficient in Eq.(10.6) is proportional to the kink density (10.4). We can relate the step stiffness fi of Eq.16.24) to the step fluctuation. In a coarsegrained continuum model, the total step free energy is written as J'step

=

J dsD (e(s)) = J dsD (d(s))

1-77 1

+

.

(10.7)

Here s denotes the arc length along the step, and O(s) is the angle between the normal vector n and y axis as tan 6' = - d y / d x . See Fig.6.4. As the step is rough, the step free energy /3 is expected to have no singularity in the orientation 6'-dependence. Thus, for a small step fluctuation, one can approximate 6'- M d y l d x , M 1 ; ( d y / d ~ ) ~p(6') , = ;(d2p/d6'2)6'2. Then the integrand in Eq.(10.7) can be expanded up to the second order of the derivative d y / d x as

+

+

d

m

(10.8) where fi zz P+P" is the step stiffness in 8 = 0 direction 160,611. From this deformation free energy Fstcp, the thermal expectation of the slope fluctuation is easily calculated by the equipartition as ( ( d y l d x ) ' } = lceT/,8. Since the slope at different positions

Part 11. Statistical Mechanics of Surface

34

are uncorrelated, the height difference correlation function G ( x ) is easily calculated

as (10.9) Comparison of Eq.10.9) with the previous microscopic result (10.6) indicates that the stiffness is related to the height difference fluctuation as

T kBT 1 + 2 c ~ p ( - J , / k ~ T ) p=- ~ B - 2a

a(.”)

exp(-J,/kBT)

(10.10)

’

We choose the zero of the y coordinate to thc average position of the step as (y(z)) = 0. Then the step fluctuation is characterized by the width w defined by

w 2 = ( ~ ( 5 ) In ~ )order . to calculate w, it is convenient to transform the step fluctuation in Fourier modes: (10.11) = ?/(4)e’”j

c q

where q = 27rm/L with the integer m = -00, * . . ,00. Here we assume the periodic boundary condition: y(0) = y ( L ) . The mode amplitude is obtained by y(g)

The step free energy

1

=-

J I’ y(x)c-iq”dz= y*(-q).

L o

(10.12)

rqtrp is written in terms of the Fourier modes as (10.13)

The thermal average of the fluctuation of each mode is obtained from cquipartition

as (10.14)

and the step fluctuation width w is calculated as

Reflecting the rougliriess of the step, its width w diverges for a long stcp as L’I2.

Exercise 1: Show that a step pinned at both ends, y(0) = y(L) = 0, fluctuates twice larger than a periodic step, and lias the step fluctuation (10.16)

10. Step Fluctuation in Equilibrium

35

The formula can be used to analyze the fluctuation of an isolated step shown in Fig.lO.3b t o obtain the step stiffness fi [4]. Answer: Step fluctuation is decomposed into sine-modes as Y(X) =

c

(10.17)

Y(Q) sinqx

Q

with q = ?rm/L and m is the positive integer as m = 1,2, is obtained by v(4) =

2

1

L

+

. . ,oo.The amplitude y ( q )

v(x)sinqxdx.

(10.18)

JL

( 10.19)

From the ortho-normal condition J L sin qx sin q'xdx = cos qx cos q'xdx = bqqt, L o L o the step free energy is written as

(10.20) From the equipartition, the thermal average of the amplitude fluctuation is obtained as (10.21)

The step width w is then calculated as

w2 =

1J L (y(x)2)dx = -1 C(Y(d2) L o

2 ,

(10.22)

Exercise 2: If the tilting field Ht is applied on the step model (lO.l), the Hamiltonian is written as

and the step is tilted such that the height at the right end ( g ( L / a ) )differs from the fixed height at the left end y(0). (1) Calculate the free energy

Q as a function of Ht

(2) Calculate the average slope p = ( y ( L / a )- y(O))/L = ( n ) as a function of H i , and show that for small Ht slope p is proportional to Hias p = Ht 2 (10.24) Pa' with the step stiffness obtained by Eq.(10.10). The stiff step resists tilting against the tilting field.

Part 11. Statistical Mechanics of Surface

36

(3) Show that the free energy of a tilted step with a slope p defined by the Legendre transformation F ( p ) = F ( H t ) LH,p/a behaves as

+

for small slope p . Answer: (1) By defining n, = (y(i) - y(i - 1)) / a , the height difference at both ends is represented as y ( L / a ) - y(0) = The partition function is calculated as

axt.=/lni.

Lfa

[1+2exp(-&)~osh~]

= cxp(-%)

The free energy

.

(10.26)

E is thcn obtained as

P = -kBT In 2 = a

(10.27)

(2) The average of the height difference is calculatcd as

( g ( L / a )- y(0)) = -a-

aF dH1

=L

2&= /'HATsinh(Ht/k BT)

1

+ 2e-J=/k@Tcosh(Hl/k~T)'

(10.28)

The slope is then obtained as 2 ~ - ~ ~ / ~ H ~ s i n h ( H , /xk ~HTI) ~ c - ~ = / ~B Ht ~ = 1 + 2 e - J = / k ~ T c o s h ( H f / k ~ T )k B T 1 + 2 C - J z f k B T )a'

(10.29)

(3) Since Lp = - a d F ( H t ) / d H f , the free energy F ( p ) is calculated as

(10.30) In the last cquality we used the approximation (10.29) for small p and the relation F " (0 ) = - ( L / a ) d p / d H f = -L/ga2. The result (10.30) agrees with the free energy of a step with a constant slope p = dy/dx calculated from Eq.(10.8).

11. Step-Step Interaction Induced by Entropy

37

Exercise 3: Consider the simih.r lattice model for the step but the step height difference between the neighboring sites, n = (y(i + 1) - y(z))/a, can take arbitrary integer values from -m to 00. Then calculate the step free energy and the roughening temperature. This roughening temperature agrees in fact with an exact transition temperature of a square Ising ferromagnet. Also calculate the height difference correlation function (n2)and the step stiffness Answer: The partition function is calculated as

p.

where the parameters ax and ay are defined as (10.32) Then the step free energy per length

00is obtained as

- 1nZ = m--= Poa

a x

ay- lncoth--.

2

(10.33)

At the roughening point, PO = 0 and thus

a y ,=~lncoth-.a x R 2

(10.34)

It is straightforward to transform (10.33) in a symmetric form as sinhax,Rsinhay,R= 1.

(10.35)

This is the relation to give the transition temperature of the square king ferromagnet with the nearest neighbor interactions, J, and Jy in x and y directions respectively [193].For an equal coupling, J, = JY = J , the transition temperature is at TR = J / k e l n ( d l ) , in agreement with Eq.(6.34). Fluctuation of a step is calculated as

+

(10.36) and the stiffness as

kBT

p=--I

a($) -

2 k ~ T pax sinh -. a 2

(10.37)

Part II. Statistical Mechanics of Surface

38

x Figure 11.1: (a) Vicinal face and (b) the top view of step configuration.

11

Step-Step Interaction induced by Entropy

Consider now a snrfacc with a small inclination, which is called a vicinal face. A vicinal facc consists of terraces of a low indexed singular facc and steps separating Wc consider for simplicity that steps arc these tcrraces, as shown in Fig.ll.1. running in 3: dircction with a mutual separation e. The slope of this vicind face is then laz/ayl = a / ! , whcre a is the step hcight. The surface normal makes an anglc 4 with z-axis: I$[ = arctan-a x -a (11.1)

e

e

for a small tilting. In a unit projected area in sy-planc, there are l / e steps running and the surfxc frcc energy f is written as (11.2)

wherc y($) is the surfacc free energy per unit area, 70is that of a flat tcrracc, and p($) is the step frcc energy per lcngth for a vicinal facc with a tilting 4. At the absolute zero tcmpcrature, T = 0, tlic step runs straight and pa is equal to Jzl of the energy of the broken bond. At il finite temperature, a single step fluctuates and the step free encrgy decreases as Eq.(10.3). For a vicinal facc, there are many other steps and they cannot cross with each other because of the large energy cost for a configuration with overhangs. Thus thc fluctuation of a step is limited by the ncighboring steps, and the ciitropy is reduced, or the step free energy is enhanced. We consider the samc step model proposed in the previous section. Since the kink formation costs an additional energy J,, thc probability that a step meanders plus or minus y dircction is equal to thc Boltzmann weight, (11.3)

11. Step-Step Interaction Induced by Entropy

39

Then meandering of a step in y-direction can be regarded as a trajectory of a one dimensional Brownian motion with a “time” x/a:At each time step Brownian particle meanders to plus or minus in y directions with a probability p~ and remains there with a probability 1 - 2pK. Then after x/a time steps, particle deviates from the original position about ( A ~ l a ) ~( x / a ) p K . If the step fluctuates a separation 1, then it collides with the other steps on average. The collision takes place at every length of Ax ap;;’(C/a)2. This means in a unit length there are (Ax)-l collisions, and the step looses the entropy. The step free energy increases by about kBT. Thus the step free energy P($J) per unit length can be estimated as N

-

a

P(d) = Po + CkBTpK-P

ckBT

exp( - J , / k B T )

Po + P a 1 + 2 exp( - J z / k B T )

Po + P242, (11.4)

where PO is the step free energy of an isolated step obtained in the previous section, and the second term represents the contribution from the step interaction with c being some constant. The coefficient P 2 is inversely proportional to the stiffness of an isolated step as P2 0: ( k ~ T ) ~ / f i ( O It )reflects . the fact that the step interaction is brought about by the step fluctuation.

Exercise:(Gruber-Mullins model [72]) In order to treat the entropy reduction by the step confinement more quantitatively, we consider a step confined between two walls. For simplicity we consider every length in unit of lattice constant a here. The wall is located at y = 0 and y = 21, so that the step height g ( i ) at the site with the a-coordinate i can take values only from 1 to 2C - 1. The system has the interaction Hamiltonian (10.1). The partition function is transformed in terms of the transfer matrix f as

(11.5) Here Tr means the trace of the matrix, and T is the symmetric (2C - 1) x ( 2 C - 1) transfer matrix. (1) Show that

? is explicitly written as

[1

1 B 0 0 B l B O

T(h,h’)=

.

!]

B f

B

.

O B 1

+

by using the abbreviation of the Boltzmann factor as B = e--’,ikBT.

(11.6)

Part II. Statistical Mechanics of Surface

40

(2) If one knows thc cigcnvalues calculated as

fib

(k = 1

N

c

21 - 1) of

T,the

trace is easily

21- I

TrTI, =

(11.7)

k&.

k=1

For a long step, L + 00, only the lasgcst cigcnvalue contribution, and the free energy F is obtained by

,&ti gives

the dominant

F =-~BT 1112 = L [J!, - kBTln p,].

(11.8)

By denoting the cigcnvectors as 4) = ($( l ) , $(2), . . ,74(2e - l), explicitly write down the cigcnvaluc equation Ttj = p$. (3) Sincc the cigcnvaluc cquation corrcsponds to thc wave cquation with a fixcd boundary condition, the eigcnvcctor $b is writtcn as

akm $k(rn) = sin -

ze

( k = 1...2e-1) .

(11.9)

Find tlic corrcsponding cigcnvaluc , u k .

(4) Find the step frec energy per length a(4)of a vicinal face with an angle (ll.l), and show that for smdl $r or large &, D(4) can be expanded as

P(4) = Do + a 2 & .

(11.10)

Answer: (2) For m = 1 . . . (2&- l ) , cigenvalue equations are writtcn as

B [$(m+ 1) - 2 + ( ~ t + ) +(rt2 - l)]3- ( 2 B + 1 - /~)l/l(m)= 0,

(11.11)

and $(O) = $(at)= 0. (3) From the simple trigonometry, one obtains the cigcnvalucs ilk

for k = 1

=1

xk + 2Bcos 2&

(11.12)

2e - 1

(4) Step free cncrgy is calculated as

=

h+P2$2,

(11.13)

11. Step-Step Interaction Induced by Entropy

41

where Po is given in Eq.(10.3) and P2

r2kBT 2exp(-J,/kBT) - r2(kBT)2 8a 1 2exp(-J,/k~T) - 8a2p(0) '

=-

+

(11.14)

In the last equation, the lattice parameter a is explicitly written again. If the neighboring steps are allowed to fluctuate, the exact calculation shows [3] that P2

=

r2(kBT)2 6a2& 0) '

(11.15)

Exercise 2: (1) Show that when the height of the vicinal face deviates E(z,u) from the mean position, the surface free energy is described by [146]

(11.16) with two surface stiffness constants 7.L

-to

= P-

a2

(11.17)

and (11.18)

p

Here is the stiffness of the steps running on average in z direction with the separation l o in 9 direction. a is the jump in the terrace height at the step. (2) Show that the width of a single step fluctuation is proportional to the logarithm of the system size L as w2

= (h2j =

-2%7T.-2 In L ,

(11.19)

by using the surface free energy (11.16),

Answer: (1) When steps are straight and aligned regularly as in Fig.ll.2(a), the vicinal face I(z,y) = 0 is tilted from the terrace face by an angle 40 = arctan(a/eo). The step fluctuation induces height fluctuation, and we consider two independent modes of step fluctuation as follows.

Let every step deforms locally at a position z in direction by h ( z ) , while keeping the step separation lo constant, as shown in Fig.ll.2(b). This step

Part 11. Statistical Mechanics of Surface

42

le Figure 11.2: (a) Flat vicinal face, (b) step bending deformation, and (c) step dilatation. deformation corresponds to the height increment Z(x,y) = h(x)a/&. For a step segment of length dx, the energy increase is given by

As(; 2 )

2

dx

(11.20)

p.

with the step stiffness In the interval dy in y direction, there are d y / & steps and the energy increase in an area element dxdy is obtained as (11.21) where (11.22) The second mode of step deformation is the compression or dilatation of the step train, as shown in Fig.ll.2(c). When the straight steps shift in y direction and one of the terrace width increases from 10 to l, the tilt angle of the vicinal face therc in y direction changes by an angle 64 = arctan(aZ/ay). This portion is inclined from the terrace face by an angle 4 = arctan(a/e). Three angles 4,40, 64 are related as 40 = 4 64, and for a small tilt relations are reduced

+

t,n

(11.23) Now we calculatc the energy change caused by the step compression. As is explained in the main text of the section, the step collision induces the energy

11. Step-Step Interaction Induced by Entropy

43

increase. For a step separation l o , the collision takes place along the step for every length of L, determined from the relation lt = kBTL,/p. The energy increase by each collision is of the thermal energy kBT. In an area dxdy, there are dull0 steps and each step makes d x / L , collisions, and thus the total energy increase is proportional to (11.24) The precise calculation [3, 1611 shows that the energy increase is r 2 / 6 times larger than the above estimation. Therefore, the terrace width variation leads the energy increase of

Since the average orientation of the whole vicinal face is fixed,

iL"

$dg = ,z(Ly)- E(0) = 0

(11.26)

and the linear term in the slope d.z/dy is irrelevant in the free energy (11.25). By neglecting the third order term of the slope in (11.25), we get (11.27) with (11.28) By summing up two contributions, (11.21) and (11.27), we get the Hamiltonian (11.16). Note the strong anisotropy of the surface stiffness. For a small tilt angle ;Vl diverges whereas 711vanishes. This drastic difference of two surface stiffness constants is observed recently in He [158]. The product ylqll remains finite though: (11.29) (2) From the free energy (11.16), the Fourier-transformed height Z ( q 2 , q y ) has the

correlation (11.30)

44

Part 11. Statistical Mechanics of Surface

The height-difference correlation function of the vicinal face is then calculated

as

(11.31) Since the height z(z,O) is related to the step deformation h ( s ) as 2(z,O) = h(x)a/&, the correlation function of the step deformation is shown to diverge for large separation along the step as (11.32)

One can also calculated the height fluctuation at a point r for a finite system of size L2 as ( z ( r ) 2= ) kBT \/,lnL = Za21nL. a2 ( 11.33) 277 Y1711

This result is translated in the step width as

(1 1.34) The width of a step in a step train diverges weaker than that of an isolated step due to the confinement effect of neighboring steps.

12

Faceting Transition

We now study the effect of surface roughening on the crystal shape in equilibrium. The equilibrium shape is determined by Andreev's formulation as Eqs.(6.8) and (6.11). Therefore, once the free energy f is known, the equilibrium shape cm be calculated. Below the roughening temperature 2'8, the steps on a vicinal surface with an inclination angle 4 in 3-direction (Fig. 11 .la) has the step free energy (11.4) or

Pt4) = Po 3. P d 2 .

(12.1)

Then the surface free energy per ~9 projected unit area, (11.2), is represented as (12.2)

Here we have used thc symmetry consideration that the surface with positive and negative tilt have the same density of steps and should have the same f . Thus f has a cusp singularity at 4 = 0 as shown in Fig.12.la. The vicinal face with ;Islope

12. Faceting 'lkmsition

45

Figure 12.1: (a) Surface free energy f per zy projected area, and (b) the faceted equilibrium shape near y = 0 below the roughening temperature TR.

p , = &/ay = t a n 4 M 4 is located at a distance y apart from the center, where y is given by Q. (6.8) as (12.3)

Here sgn(4) represents the sign of the slope 4. Since A, PO and P 2 are positive for T < TR,the slope is positive for y < 0 and negative for 3 > 0. Let us consider the region of y > 0 and d, < 0. (See Fig.12.1.) Then Eq.112.3) is solved for d, as (12.4)

for g larger than yo = Po/Xa. The height z is determined from Eq.(6.11) as

Az = f + Ayp,

= 70 - 2a-'P21+13.

(12.5)

The crystal shape is then written a9 (12.6)

for y > 90. Here 20 = ?o/X. For y < -yo, d, is positive and the profile is similar to Eq.(12.6) with only a replacement of y to -y. Therefore, between -yo and yo the shape consists of a facet face, and then it continues to the smooth curved faces with the exponent 3/2, ;t9 shown in Fig.12.lb [l60]. The facet size 2y0 is proportional to the step free energy density, PO. Neaz the roughening point TR,PO and thus 90 vanish smoothly as yo e x p [ - c / d F q . This extinction of the facet face is called the faceting phase transition, and is equivalent to the roughening transition of the face. There are experimental studies on faceting transitions of 4He crystals in superfluid liquid. The transition temperature is determined to be 7'~=1.3K for (0001) face. The roughening temperatures of other faces are also determined [68]. N

Part I11

Kinetics-Limited Growth In this part, the growth laws governed by the surface kinetics on a flat crystal surfaces are studied. When the crystal grows with a flat surface, the growth is controlled by the incorporation of atoms to steps and kinks provided by the two-dimensional nuclei [12] or screw dislocations [62]. Since the step density depends on the growth condition, thc growth law is different from the ideal ones.

13

Step Advancement by Surface Diffusion

We consider the crystal growth from the vapor phase at a relatively low temperaturc! where a crystal surface is flat. Atoms deposited on the surface diffuse around on the surface before they reach steps. The advancement of a step in thc surface diffusion field of adsorbed atoms is studied systematically by Burton, Cabrera and Frank [43]. We assume that the adsorption sitc on a crystal surface forms a square lattice with a lattice constant a. From the amhicnt gas phase f atoms deposit on a surface per unit area in a unit timc intcrval. Therefore, on a lattice sitc (z, y) with atomic area R2 = a2, &At atoms are dcpositcd in a time interval At. The adsorbed atom vibrates with a frequency v of the lattice vibration around the averagcd adsorption site, and tries evaporation or surface diffusion. Due to the coupling with tlic underlying crystal atoms, the adatom has an energy lower than in the frcc state by E d , and thc evaporation takes place with the probability exp( -E,,/knT) during I / trials per unit time. The lifetime until the evaporation is estimated as

(13.1)

r = v-' e x p ( ~ d / ~ ; B ~ ) .

By denoting the adatom density as c ( x ,y), the probability that a lattice sitc (z, y) is occupied by an adatom at a timc t is c ( x ,y; t)&. The number of adatoms cvaporating within a time intcrval At is given by c(x,y; t)R2. At/r. On the other hand, an adatom hops to one of thc neighboring sites during its lifctimc. For the random walk of the adatom from one adsorption sitc to the other it has to cross over the activation energy or energy barrier Esd, and the probability of hopping to the neighboring sitc is w = vexp(-E,,l/knT). The number of atoms coming from the four neighboring sites during the timc At is wAtR:![c(x a, y;t ) c(x - a, y; t )+ c ( x ,y + a; t ) + c(z, y - a; t)], while the number of atoms escaping from the sitc (x,y) is wAtR24c(x, y; t ) . Combining all these processes, thc variation of the number of averaged atoms at the site (x,y) during the time interval At is written as

+

C(Z,

9; t

+ At)R2 = ~ ( 59;, t)R2 + fR2At

- ~ ( 5y;,

+ w02At[c(x+ a, y; t ) +c(x - a, yj; t )+c(z, y+a; t ) + c ( x ,y -

+

At r

t)R2-

a; t ) - 4c(x, y; t ) ] . (13.2)

47

13. Step Advancement by Surface Diffusion

After dividing Eq(13.2) by RzAt, take the continuum limit A t + 0 and a + 0 and leave up to the second derivative of the space. Then one obtains the diffusion equation including the contribution of deposition flux f and the desorption lifetime as (13.3)

Here

D,= wa2 = va2exp(-E,d/kBT)

(13.4)

is the surface diffusivity. How far can the adsorbed atom meander while it resides on the surface? During the time t , an adatom makes wt jumps on average. Since it can jump to four nearest neighbors with the same probability, the average position (r(t)) remains the same with the original one, (r(0)). But the deviation ((r(t) - r(0))2) increases in proportional to the number of jumps. Since it hops a distance n in a single jump, the deviation is written as ((r(t) - r(0))') = wtn2 = D,t. Therefore, during its lifetime 7 on the surface, an adatom meanders in the range x, = d D S r ,

(13.5)

which is called mean displacement of the adsorbed atom or the surface diffusion length. During the surface diffusion an adatom reaches a step and then crystallizes by being incorporated in the kink sites. Let us consider the advancement of an isolated step running on average in the x-direction. Usually the step advances so slowly compared to the relaxation of the adatom density c that the stationary distribution of c can be realized: The time derivative in Eq.(13.3) is disregarded. If the step is straight in the x-direction, the concentration variation takes place only in y-direction, and the diffusion equation is simplified as (13.6) where coo = f r is the concentration far from the step. The solution is easily obtained a3

c - c, = A exp

(-$)

(13.7)

with the surface diffusion length 5,. Since the step is rough at finite temperatures as shown in section 10, the kinetics of adatom incorporation is expected to be very fast. In the extreme case, the step can grow with only an infinitesimal driving as c(0) = cFq. From this local equilibrium boundary condition at the step, the integration constant A is determined to be A = ceq- c,, and the diffusion field is obtained as c(x,g;t) = c, - (c, - c,)exp

(13.8)

48

Part 111. Kinetic-Limited Growth

C

........... ...... ....T'-""'

.....".,.

7 ~

XS

Y

Figure 13.1: Concentration distribution around the step at y = 0.

as is depicted in Fig.13.1. The net number of atoms impinging to the step at y = 0 from the right (y > 0) is 2002 [c(z, a; t ) - c(x,O;t ) ] = wa3 &(x, ~ ) / L ? y l ~ = +and ~ , that from the left (y < 0) is wf& [c(z, -a; t ) - c(z, 0;t ) ] = -wa3 dc(z, y)/d~l~,-~.Here y = +O(-0) means in front ( in the back ) of the step. Since these deposited adatoms crystallize at a kink site on a step, and step advances a distance a by each crystallization, the advancement rate of a straight step is obtained as

By inserting the density profilc (13.8), the step velocity is obtained as

Here f e s = ceq/7- is the equilibrium deposition flux. The result (13.10) can be interpreted as follows: The adsorbed atoms landed on a surface within a distance of their mean displacement 5, from a step will eventually meet the step during their lifetime, and be incorporated to it. Therefore, f x 25,a atoms impinged in a surface region of a stcp length a within a distance 2, from the stcp will crystallize at a step in a unit time. As an inverse process, fc, x 2x,a atoms melt back from the stcp onto thc surface. Since each crystallized atom pushes the step ahead by a distance a , the advance rate is given as vo = (f25,a - f,,2s,a) x a in agreement with the result of the detailed calculation (13.10). So far we considered an isolated step on a singular flat surface. When the surfacc is tilted with a small angle, there arc many steps arranged periodically: The vicinal surface consists of a step train. When the separation C between periodic steps is smaller than the diffusion length a:*, the diffusion ficld of neighboring steps overlaps and competes. A single step gets a contribution only from the small range around itself with a width e. Therefore the advance velocity will decrease to v = (f fes)a21 = vo x (C/22,). Of coursc, when the step separation C is wider than thc

14. Advancement of a Circular Step

49

surface diffusion length x., each step advances with a velocity vo as if it is an isolated single step. In a more precise treatment, the solution of the diffusion equation (13.3) is written as c(y) - cm = Aexp(-y/x,) Bexp(y/xc,)within a region 0 < y < I. By taking into account the local equilibrium boundary conditions at neighboring steps, c(y = 0) = c(y = I ) = G ~the , integration constants A and B are determined, and the concentration is obtained as

+

c(x,y; t ) = coo - (c,

cosh [(2y - I)/2x,] cosh(l/2x,)

- ceq)

(13.11) '

From the material conservation (13.9), the advance velocity of a step is given as

v = (f - feq)a22x.tanh

(&)

= vo tanh

(&)

.

(13.12)

If the step separation I is much larger than the diffusion length x,, the advance rate v is equal to the isolated step vo in (13.10). If I is smaller than x8,then the approximation tanh(.t/2xs) M I / 2 x , holds and the velocity is approximated as v M (f - feq)a2t, as already stated.

Exercise: Derive Eq.(13.12). Answer: By imposing a local equilibrium boundary conditions at y = 0 and y = I, one gets

+ B = -,C Ae-e/xe + Bee/". = ccq -,,c

~ ( 0-),C

= A

,$I) -,C

=

ceq

(13.13)

and the integral constants are obtained as ee~2x.

A = (ceq ), '

2cosh(I/2xc,)

(13.14)

and (13.15) By putting these results, one gets (13.11). From Eq.(13.9), we get the velocity (13.12).

14 Advancement of a Circular Step When a two-dimensional crystal nucleus grows in a diffusion field, the encircling step expands and the total step free energy increases. In order for the nucleus to achieve further expansion, this energy cost has to be overcome. How will then the step advancement rate be modified from the straight step?

50

Part III. Kinetic-Limited Growth

The chemical potential of the bulk crystal ps agrees with that of the adsorbcd atoms with the density c, in equilibrium: ps = pd(ceq). If a circular nucleus with a radius p is in equilibrium with adatoms, the radius corresponds to the critical one since it does not grow or shrink. Then the chemical potential p&(ccq(p)) for adatom with density c,,(p) should satisfy thc critical condition (6.14): A p = 1id(ceq(p))- ps = pflz/p. Here the step free energy p is assumed isotropic. Since thc ideal solution approximation (5.2) is applicable for a dilute adatom system, the chcmical potential difference is writtcn in tcrms of adatom density as

The last approximation is valid for a step with a small curvature IC = l/p. The curved step needs a higher equilibrium concentration than the straight step does in order to remain in equilibrium. The effect is called the Gibbs-Thomson effect. The adatom density far from the nucleus is determined by the deposition flux as c, = fr. Whcn the density c, is higher than the equilibrium value h ( p ) , thc circular nucleus with a radius p grows. The density distribution around the circular nucleus is expected to be symmctric and to depend only on the radius T . The diffusion equation is simplified as (14.2)

For the slow growth, the stationary approximation &/at = 0 is expected to hold. The exact solution of c is described in Appendix A14. Hcre I derive the circular step advancement in approximation. The radius of the nucleus p is assumed large. Since the step advancement is controlled by the diffusion field around the step, the radial variable T in Eq.(14.2) is also large, and the second term, r-ldc/ar, may be neglected compared to the first term, d2c/dr2.The stationary density distribution is obtained as (14.3) with 2, = being the surface diffusion length defincd in Eq.il3.5). The adatom density relaxes to c, far outside (r >> p ) and far inside (r << p ) of the cluster. By imposing thc local equilibrium condition c(p) = c,,(p) at the circular step, the integral constant A* is dctcrmincd as A+ = ceq(p)- cm. Then the step advancement velocity in the radial direction is obtained as

(14.4)

15. Growth Rate by Spiral Growth Mechanism

51

where vo = 2DsR2(c, - ~ q ) / xiss the velocity of the straight s k p (13.10), and (14.5) is the critical radius corresponding to the deposition flux f or Eq.(6.14). Therefore, the nucleus with critical radius pc does not grow or shrink, the one with radius p > pc grows, and the one with p < pc shrinks. The correct calculation in Appendix A14 leads to the same conclusion.

15 Growth Rate by Spiral Growth Mechanism The growth rate V of a crystal by spiral growth mechanism is now ready to be calculated. The step is running out from the center of the screw dislocation. As the crystallization proceeds, the initially straight step turns round the dislocation, and the curvature at the center increases, as shown in Fig.lO.1. But it should be smaller than the critical one, K~ = l/pc. Otherwise, the crystal melts back due to the Gibbs-Thomson effect. Therefore, the curvature at the center is just the critical one in the steady state. The simplest form of the spiral is that of Archimedes, represented in the two-dimensional polar coordinate (r,r$)as r = a4 with a to be determined. As is explained in the last paragraph of Appendix A15, the curvature at r$ -+ 0 is calculated to be K = 2 / a . Therefore, the parameter a should be 2pc in order to satisfy the boundary condition. The step separation e for large r is given as the radius difference by one turn, or A4 = 21r rotation as

t = a * 2~ = 4np,.

(15.1)

The more sophisticated calculation is described in Appendix A15 [44]. It shows that the asymptotic step separation for large r is

e x igp,.

(15.2)

-

In both cases, step separation t increases in inversely proportional to the driving force, t Ap-l. Since the advancement velocity v ( t ) of a step train is given by Eq.(13.2), the time T required for the step to wind up one turn is given by e/v(e). During this time T the crystal surface grows a height a in the normal direction, and its growth rate of the crystal is then given as a V = v(e)- = a3(f -

e

(15.3)

and the dependence on the driving force Ap is shown in Fig. 15.1. For a low vapor pressure p , the step separation f2 is larger than the surface diffusion length za, and tanh(C/2z8) x 1. The growth rate is approximately given by V =

52

Part 111. Kinetic-Limited Growth

Figure 15.1: Growth velocity V versus driving force u = exp(Ap/kBT) - 1 for the spiral growth.

a3(f - feq)22./l proportional to the square of the driving force ( p - P,,)~ N AIL'. Thus, at low vapor pressure growth follows thc parabolic law, different from the linear growth law of Hertz-Knudscn. When the vapor pressure p increases, the driving force for the crystal growth A p becomes large, and the critical radius pc and the step separation t becomes small. For << G, the approximation tanh(l/2x8) M l?/22, holds, and the growth rate reduces to the Hertz-Knudsen formula given in Eq.(4.4): VHK= a3(f - feq) = a 3 ( p - pes)/d=: There are many steps and kinks on the surface, and the growth rate V reduces to that of a rough surface.

e

16

Two-Dimensional Nucleation and Growth

Spiral mechanism governs the growth of an imperfect crystal under a small driving force. When the supersaturation increases, nucleation becomes important. Also for a perfect crystal without dislocation, nucleation controls the crystal growth. In this section thc growth rate of a crystal by nucleation mechanism is studied f l 2 , 80, 691. The formation of a two-dimensional (2D) nucleus increases the free energy as

- Apnp2 + 2app = - A p . n +a{= 02

= G(n),

(16.1)

where p is the radius of the 2D nucleus, Ap = p~ - cis the chemical potential gain by the crystallization, 522 the arca of a nucleating atom, /3 the step free energy per length and assumed to be isotropic hcre. There are n = 7rp2/R2 atoms in the circular nucleus of radius p. G(n)has a maximum as shown in Fig.16.1 at thc critical nucleus size 11,: (16.2)

16. Two-Dimensional Nucleation and Growth

53

Figure 16.1: Free energy cost G(n)of the formation of the two-dimensional circular nucleus with the size n.

0

+o

Figure 16.2: Variation of the cluster size n by capturing or by evaporating a single adatom. The maximum value of the free energy barrier is

G, = G(n,) = A,un,

=

@Rz -

(16.3)

AP Let us consider the distribution C ( n , t )of the cluster of a size n at time t . In equilibrium thermal fluctuation allows the formation of nucleus of a cluster size n with the Boltzmann weight Co(n)= cmexp(-G(n)/lc~T), where c, is the density . that the equilibrium density of infinitely large of isolated adatoms: c, = f ~ Note nucleus is infinite. During the steady growth of a crystal, in fact, a large nucleus disappears by the completion of a layer and the density of an infinitely large cluster in a steady state should be zero C ( n 00) = 0. The size variation of the nucleus is assumed to take place by capturing or cvaporating a single adatom as shown in Fig.16.2. The rate equation is written as --f

--

at

- w+(n- l ) C ( n - l)+w-(n+l)C(n+l)-zu+(n)C(n)-w-(n)C(n),(16.4)

where w+(n)is the rate of capture of a single adatom by the nucleus with the size n and n ( n ) is the rate of evaporation of an atom from the nucleus of size n. The detailed balance condition

w+(n)CO(n) = w-(n

+ 1 ) c o ( n+ 1)

(16.5)

Part III. Kinetic-Limited Growth

54

confirms the approach to equilibrium. Then the ratio of the capture and evaporation rates is obtained as

+

+

exp[-G(n l)/k~T] w+(n) - Co(n 1) w-(n+ 1) - CO(n) exp[-G(n)/bT] '

(16.6)

In a steady state, X ( n ) / a t = 0 but there is a net flow J of the cluster growth supported by the constant supply of particles from the smallest size and their annihilation at the largest size. J is then independent of the cluster size as

J = w+(n)C(n)- w-(n

+ l)C(n + 1) = w+(n)CO(n) ) ; : : [ --

:n,f+y

.

(16.7)

For n = 1 there are plenty of isolated adatoms and one assumes that C(1) = Co(l). As for the boundary condition at n = 00, the infinitely large nucleus disappears by the completion of a monatomic laycr and thus C(m) = 0. Then by using the relation

J

- 1,

(16.8)

the nucleation flux J is obtained as (16.9) Since C0(n)-' = c i l exp[G(n)/kBT) has the maximum at the critical nucleus size n, of Eq.( 16.2) and w+(n)varies slowly in proportional to the surface area as w+(n)N n'f2, the dominant contribution to the summation in Eq.(16.9) comes near n,. For small A p , n, is so large that the summation can be replaced by the integration as

Here G, is the maximum of the free energy cost given in Eq.(16.3) and G(2)is the second derivative of the activation free energy at n, as (16.11) Therefore, the nudeation rate is written as J x Zw+(n,)c, exp ( - G c / k ~ T )

(16.12)

16. Two-Dimensional Nucleation and Growth

55

with the Zeldovich factor

Since the adatom capture by the nucleus takes place at the periphery of the nucleus, w+(n,) is proportional to the perimeter length as [80, 691 (16.14)

where VO,+ = 2zsQ2f is the step advance rate by atom incorporation in Eq.113.10). If there is a direct incorporation of atoms from the ambient gas phase, this contribution should be included in vo,+ [69]. Thus (16.15) When the nucleation rate J is large, nucleation starts at various places on a crystal surface and those nuclei spread with the advance velocity v, coalesce and complete a single layer. This mode of layer growth is called the multinucleation growth. We now calculate the normal growth rate V of the crystal in this situation. On a flat surface it takes a time T for the completion of a single layer growth. During this time interval T, nucleation occurs with the rate J per unit area and time. At a time t after the nucleation started, its radius is vt and occupies the area n(vtj2. Therefore, during T the area scanned by nuclei is calculated as

~ T ( S J ) r ( u ~= ) 2S-u dK J~ 2 T3 , 3

(16.16)

where S is the total area. When the scanned area (16.16) coincides with the total area S, the single layer growth is completed, and thus the time T is determined by -rvJT2 3 --1 .

3

(16.17)

After this time T the crystal surface moves up a distance a, and the normal growth rate by the two-dimensional nucleation mechanism is given by

~ = a - = raJ v 2( ~lI3 ) T

(16.18)

Since the advance rate of the step is given in Eq.(13.10) or (16.19)

Part III. Kinetic-Limited Growth

56

. V

f/fq - 1 = exp(Ap&r)

-1

Figure 16.3: Scheinatic diagram of the growth velocity V by tlie two-dimensional nucleation. the normal growth rate is

The last approximation is obtaiiied for small supersaturation Ail wliere ,c = ,,c, and the result is schematically sliown in Fig.lG.3. Since V is proportional to tlie exponential of Ap-', it is very m a l l for small ALL.The rate V becomes observable only when Gc(Ap) is of t,he ordcr or slnaller than tlie thermal energy ~ B T ; (16.21)

-

For large enougli Ap, tlie nucleatioii harrier vanishes, and the growth rate is V (Ap)'/6exp(4Ap/3koT), larger than the linear growtli law V cx e A J 1 / k B T - 1. This result is incorrect. For large AIL,the critical nuclear size n, becomes so small that n, < 1, and the crystal surface 1)ccoiiies kinetically rough. Then tlie growth rate should be that of the rough srirfacc or tlie Hertz and Kiiudsen formula (3.4). At the thermal rouglieiiiiig teiiiperature TR and above, tlie step free energy /3 vanishes, and tlie energy barrier G, disappears. Tlius the growth mode varies from the nucleation controlled exponential type, V (:xi,( -C/Ap) below the roughening A p above Tn. This is transition temperature TI<,to tlic nornial growtli rate 1' used for the criterion to dcterniinc the rouglieniiig temperature in the Monte Carlo

-

-

16. Two-Dimensional Nucleation and Growth

57

0.4

2 L 2

0.2

0 Ap/h

Figure 16.4: Growth rate R versus chemical potential difference of the SOS model. Numbers aside the curves represent inverse temperatures, 6 J l k ~ T .k+ here is the deposition rate, and d is the atomic height. The dashed lines are the growth rates calculated from the two-dimensional nucleation model [70]

HEIGHT H(mn f

(4

I/H (ma”)

fb)

Figure 16.5: Growth rate of He crystal near and below the roughening temperature TR= 1.23K (a) in the normal and (b) in the semi-logarithmic plots [198]. H here is proportional to the chemical potential difference Ap. simulation as is shown in Fig.16.4 [70], and in the experiment on He as shown in Fig.16.5 [198].

Part 111. Kinetic-Limited Growth

58

Exercise: When the nucleation rate J is very slow or the step advancement velocity v is very fast, there is only a single nucleus during the layer growth. Then the normal growth rate V differs from multinucleation case. Calculate the normal growth rate V of a surface with an area 5’. This growth mechanism is called the single nucleation growth. Answer: When a singlc nucleus is created, it spreads over the whole surface S quickly before further nuclei arc formed on the same level. The time T necessary for the completion of the normal growth of a height u satisfies thc relation JST = 1.

(16.22)

The growth rate is obtained as (16.23) If one plots In V versus l/Ap, the slope by a single nucleation mechanism is threc times larger than that by a multiple nucleation mechanism. This crossover is said to be observed in polymer crystallization 181, 1791.

17

Asymmetry in Attachment Kinetics

In the treatment of adatom diffusion, Burton, Cabrera and Frank (BCF) [43] assumcd the fast kinetics at the step such that the local equilibrium is realized there. Also they assumed the symmetry in thc attachment kinetics of adatoms from the upper and lower terraces at the step. However, the incorporation rate of an adatom from an upper terrace needs not to be same with that from a lower terrace as shown in Fig.17.1 [172, 1901: From the upper terrace an adatom has to break many chemical bonds with the underlying substrate atoms when it crosses over the step down to the crystallization position. From the lower terrace, on the other hand, an adatom

Figure 17.1: Asymmetry in the incorporation of an adatom (a) from the upper and (b) from the lower terraces, called Schwoebel effect. (c) The potential energy profile of the adsorbed atom.

17. Asymmetry in Attachment Kinetics

59

can simply make additional bonds with the step atoms before it reaches kink sites. Additiond energy barrier for the crystallization makes the kinetic coefficient from the upper terrace K- smaller than that from the lower terrace K,. This asymmetry in attachment kinetics at the step is first studied by Schwoebel and Shipsey and is called the Schwoebel effect [172].

17.1

Schwoebel Effect

We consider the step down configuration, where the step is running on average in x direction at g = 0 and the terrace in front at y > 0 is lower than the terrace in the back y < 0. Since the step is thermally rough, the step advance rates, v+ and v-, by the adatom incorporation from the upper and the lower terraces respectively are linearly proportional to the supersaturation as (17.1) Here the Gibbs-Thomson effect of curvature is included. The Schwoebel effect means that the kinetic coefficients are different: K+ # K-. We now restrict ourselves in the extreme and the simple case that

K-=O

and

K+=oo.

(17.2)

In this case, there is no crystallization from the upper back terrace. Since the crystallization takes place by the atom incorporation only from the lower terrace, it is called a one-sided model. Furthermore, the kinetics from the lower front terrace is assumed extremely fast such that the local equilibrium is realized: The adatom density in front of the step c+ is equal to the equilibrium value cq with the Gibbs-Thomson effect c+ = cq (1

+ mfin2 K)

(17.3)

As in the BCF model, the atoms are deposited on the crystal surface with a flux f , the adsorbed atoms then diffuse on the surface with a surface diffusion constant D., and then evaporates back into an ambient vapor after a life time r. For a straight step its advance velocity uo is calculated to be (17.4) is the surface diffusion where f, = c q / r is the equilibrium deposition flux, x. = length cmd 0 2 the atomic area. The velocity vo is half of that given in (13.10) in BCF theory, since there is a contribution only from the front terrace. Eq.(17.4) shows that the step incorporates atoms deposited in the range xs in front of the step. We now consider the stability of the straight step 181. If the step is pushed forward at some part by fluctuation, the region to incorporate adatoms expands radially as

Part III. Kinetic-Limited Growth

60

Figure 17.2: Capturing region of adsorbed atoms of a one-sided model in front of a step (a) for a straight step and (b) for a curved step. (c) Capturing region for a symmetric model for a curved step. is shown in Fig.17.2. The area of the capturing region increases approximately by a factor 1 xs/2p, and the velocity increases with the same factor. Thus the step with a curvature K = l / p has a velocity higher than the straight one by

+

SVd

= 210.

xs --R.

(17.5)

2

The bump is accelerate compared to the straight part and is pushed further forward. The diffusion causes a destabilization of a step profile. Competing with this destabilization is the stabilization effect by the step stiffness. Since the equilibrium density increases at a curved part, the deposition flux to maintain equilibrium at a curved step increases as feq(l ,&K/IC~T),and the driving force or the supersaturation decreases correspondingly. The velocity of a bump decreases by

+

(17.6) Both destabilizing and stabilizing effects are, in the first order approximation, proportional to the curvature, as is apparent in Eq(17.5) and (17.6). Since the diffusional instability increases with the velocity 210, the instability wins eventually by increasing the deposition rate. The instability takes place when 6ud 6v, = 0. The critical deposition rate fc is determined from the Eqs. (17.4-17.6) as

+

(17.7)

Exercise: Explain that in the BCF model where I<+ = K- = oa,the step instability does not take place.

17. Asymmetry in Attachment Kinetics

61

Answer: Consider a part of a step pushed forward with a positive radius p as shown in Fig.17.2~. The velocity contribution from the lower terrace increases as v+ = v o ( l + s./2p), but that from the upper terrace decreases as 2)- = vo( 1 - x s / 2 p ) . The total velocity v+ v- = 2v0 remains independent of the step deformation in the stationary approximation.

+

17.2

Structure of an unstable step

In order to describe the profile of a destabilized step, one has to treat the problem quantitatively and analytically. First we treat the stability of a straight step in a linear analysis [8, 1861. The step deformation y = C(z,t) is decomposed in Fourier modes. In the linear analysis there is no coupling among modes, and the consideration of a single mode with a wavenumber q is sufficient: ((z, t ) = uot

+ uqewq'cosqx.

(17.8)

Here vo is the velocity of a straight step moving in y direction. If the amplification rate wq of the deformation is negative, the amplitude of the deformation diminishes and the straight step recovers. On the contrary, if wq is positive, the mode amplitude increases and the straight step is unstable. The normal direction of this deformed st,ep (17.8) is given by (17.9)

(17.10)

and the growth velocity is obtained as v = (0, a(/&) = (0, vo

+ wqaqewq'cosqx).

(17.11)

Therefore the normal velocity is written up to the first order of a, as

v,,= (n . v) = vo + wgaqeWqf cos qx.

(17.12)

The adatom density c(x,p;t ) is also modified with a wavenumber (I in x direction. For y -+ 00, the modification should decay and thus the density is written as c(z, y;t ) = c,

+ (cq - cm)e-(~-uOf)/x~ + sc ewq'cos qxe-Aq(Y-vot) 9

(17.13)

in front of the step. The first two terms are the density distribution around the straight step. Since the density satisfies the diffusion equation in the stationary approximation, c, - c (17.14) v2c 2 = 0, x*

+

Part 111. Kinetic-Limited Growth

62

Figure 17.3: Dispersion relation of a step deformation with Schwoebel effect. the damping coefficient A, in y direction satisfics the relation -q2

A, = J-.

+ A:

- x;* = 0 or (17.15)

From the local equilibriuni condition at the step, Eq.(17.3), the density deviation 6c, (17.16) where the curvature IC is approximated in the linear approximation (17.10). k o m the continuity boundary condition (17.17) one obtains thc relation

By comparing the zeroth order term in aq,one obtains the velocity wo of the straight step (17.4). From thc first order of a,, the dispersion relation between w, and q is obtained as

= v0(Aq - xi') - D,R2A -4 %P 'ksT

2,

(17.19)

which is shown in Fig.17.3. The first term corresponds thc diffusional destabilization or 6 v d in (17.15), and thc second term corrcsponds to thc energetic stabilization or

17. Asymmetry in Attachment Kinetics

63

6v, in (17.6) for long wavelength or small q (< xs). The dispersion relation (17.19) can be further rearranged as (17.20) Wq = -pq * vq with the mobility pq = ~.fi:~,s(> 0) (17.21) and the restoring force

(17.22)

where the long wavelength limit, qx, < 1, is used in the last approximation. The effective stiffness ,&.r is the force constant for the step recovery modified by the diffusion destabilizing effect as (17.23) Here fc is the critical deposition rate defined in the previous subsection, (17.7). At the instability point fc, the effective stiffness vanishes, and the step looses the restoring force to the straight form. Equation (17.23) shows that the step stiffness depends on the deposition rate f. For small f Pefi is large and the step is stiff, while for large f it is soft. This variation of the step stiffness is observed in the experiment on Si [125, 1261.

Near the instability point

fc,

we have a small parameter (17.24)

For positive E , the dispersion relation for long wavelength has a positive maximum at qm of order as shown in Fig.17.3. The maximum value of w, is of order.'E This mean that near the instability point, the step modulation has a large spatial extension of order e-1/2, and relaxes slowly in a time of order E - ~ . In the weakly nonlinear case the dynamics of an unstable step can be analyzed by rescding the variables and extract slow dynamics [199, 131. One scales space and time as follows;

x

= &2XjX,

y = YjXa T = c2t/r,

(17.25)

and introduce a dimensionless field u and an interface position 5, which are expanded in terms of E as

u = R2(c - c),

= 210

+ + EU1

C2U2

+..

'

Part 111. Kinetic-Limited Growth

64

The diffusion equation is described in these variables as €UXX

+ U y y - u = 0.

(17.27)

Here and later in this section, the subscript X or Y means the derivative by them. The boundary conditions are written as

V,(l+

E)

u* = -K(l

+ E ~ H=T~y - E 2ThxH,y

(17.28)

V, e2Hxx +E) - 2 [1+ E ~ ( H x ) ~ ] ~ / ~ '

(17.29)

where V , = 2@ceqfi/k~Tx.is the step velocity at thc critical point each order of c, one obtains the solutions and relations as

fc,

By comparing

(17.30) (17.31) 0(c2):

A2 1 1 1 + pAl,xxYe-Y - = HO+ ZH: + pHo,xx +Hi ' A0

u2

= A2e-Y

~3

1 1 = A ~ C -- ~ A ~ , X X Y C - ~-Ai,x,yxx(Y2 + Y)c-', 2 8

(17.32)

O(f3) :

+

+

(17.33) To fulfill the boundary conditions (17.28) and (17.29) consistcntly at the ordcr Off3), HOshould satisfy the differential equation as

(17.34) The linear part rcprcsents the dispcrsion relation (17.20-17.22) in this unit. The is the lowest possible term compatiblc with the translanonlinear term (8H0laX)~ tional symmetry of the system. There cannot bc terms containing HOsuch as H i or Ho8Ho/aX since the cvolutioii should be indepcndcnt of the absolute position of the step itself. The equation (17.34) is called thc Kuramoto-Sivashinsky equation [118, 1771 and is known to have spatio-temporal chaos, as shown in Fig.17.4. The front has many hills and valleys. Vallcys shift randomly to the left or to the right, collide with each other and annihilate. A hill widely spread splits and a new vdley is formed randomly. The samc chaotic behavior is observed in the Monte Carlo simulation of the step advancing in the adatom diffusion ficld, as shown in Fig.17.5 [170].

17. Asymmetry in Attachment Kinetics

h

65

V

Figure 17.4: Time evolution of the Kuramoto-Sivashinsky equation, showing sr>atiotemporal chaotic behavior. (a) The height h and (b) thegradient = ah/&. .

7

Figure 17.5: Monte Carlo simulation of the time evolution of an unstable step growing from the diffusing adsorbed atoms 11701. It shows the spatio-temporal cham similar to Fig.17.4.

Part IV

Diffusion-Limited Growth: Pattern Formation So far we considered the effect of surface structure and surface kinetics on the crystal growth. But there are other processes which controls the growth. Crystallization proceeds in the following sequences: 1. Atoms to be crystallized are transported to the crystal surface (= chemical diffusion), 2. they are incorporated in the crystal at the surface (=surface kinetics),

3. and the released latent heat should be transported away from the crystal surface (= heat conduction). If all these processes are fast enough, ideal growth laws can be realized. But in reality, the slowest process governs the growth rate as a whole, and the deviation from the ideal linear growth laws explained in part I is expected. For the crystal with an atomically flat interface, the second, surface kinetic process is the slowest one and it controls the growth. The growth law is found to differ from the ideal one, as explained in Part 111. For rough surfaces, kinetics is fast, but still the growth is different from the ideal behavior when the transport processes (1) and (3) are slow. In this part we consider the crystal growth and its morphology, when the growth is controlled by chemical or heat diffusion processes. Diffusion field induces instability in the growing interface, as, for example, shown in the previous section 17. This diffusional instability causes variety of patterns in the growth shape of the crystal. Examples are the fractal structure of the diffusion-limited aggregation (Fig.l.3), dendrite in the melt or solution growth (Fig.l.2), lamellar structure in the cutectic growth (Fig.26.2). These topics are studied in the present part.

18

Diffusion Equation

Since the mass or energy is conserved, thc material or heat transport follows adiffusion equation [122]. To consider the problem concretely, we consider the crystal growth from the undercooled melt where the heat conduction evacuates the produced latent heat from the crystal surface. Since the melt is cooled at a temperature T, below , Gibbs free energy of the crystal per volume Gs is the melting temperature T M the lower than that of the melt GL as (3.1) or

18. Diffwion Equation

67

I

crystal

melt

Figure 18.1: Temperature profile around the flat interface of the crystal growing in the supercooled melt. where L is the latent heat per volume. This AG drives the material to crystallize. By the crystallization, however, the latent heat L released at the interface heats up the crystal surface to a temperature Ti, higher than the far field value T,. Therefore, the driving force of crystallization at the interface reduces to AGi = L(TM - Ti)/TM. For many metals or some plastic materials, the interface is rough at the melting point and thus the growth rate is proportional to AG;, following the Wilson-Frenkel formula (3.5) or V, = KT(TM - T i ) (18.2) with the kinetic coefficient KT. The remaining supercooling Ti - T, drives the heat transport and prohibits the interface from being heated up by the latent heat. The heat transport in liquid is described by the heat conduction equation

m

C-=kV

pat

2T,

(18.3)

where C, is the specific heat per volume, k is the thermal conductivity. When the crystal grows with a normal velocity V, the latent heat LV, is produced per unit area in unit time. This heat should be transported away by the heat flow in the normal direction n as LV, = -k(n. V ) T = -ka,T. (18.4) Here we restrict ourselves to the one-sided model such that the heat is transported only in the liquid. Extension to the two-sided or symmetric models is straightforward. The driving force AG = L(TM - T,)/TM is partitioned into the kinetic part AGi and the transport part AGt = L(T,- T,)/TM.The interface temperature Ti is so determined that the velocity determined from the kinetics (18.2) agrees with that from the transport (18.4). The fundamental equations (18.2-18.4) is derived for the melt growth where the heat conduction controls the growth. Similarly, in the solution chemical diffusion controls the growth of crystal. In order to generalize the situation and to take up the essential features of the problem, we scale variables in dimensionless form. The

Part IV. Diffusion-Limited Growth: Pattern Formation

68

dimensionless diffusion field is defined as (18.5)

and it follows the diffusion equation

au

- = DV2u

(18.6)

at

with the thermal diffusivity D = k/C,,. Boundary conditions are the conservation law

V, = -D&U,

(18.7)

and the Wilson-Frenkcl law at the interface

V, = K ' ( A - U,- d ~ ) ,

(18.8)

where K' = KT(L2/C,Tbf)is the new kinetic coefficient, which will be denoted K hereafter. (18.9) is the dimensionless undercooling, which is the undercooling normalized by the temperature increase caused by the latent heat production. tii = (Ti - Tm)/LC;' is thc valuc of the diffusion field at the interface, (18.10) is the capillary length proportional to the surface stiffness

7,

and

K.

is the curvature.

For the rough interface with many steps and kinks, thc kinetics is expected t o be very fast. In the limit of I< + 00, the local equilibrium is realized at the interface as U,= A

- dK,

(18.11)

instead of the Wilson-Frenkel law (18.8). In this caw, the crystal growth is totally governed by thc diffusion.

19

Flat Interface

When the crystal grows steadily with a macroscopically flat interface in z direction, the growth velocity V is expected to dcpend on thc undercooling A. In the coordinate frame comoving with the crystal (2,u, z' = z - V t ) , the diffusion equation (18.6) is transformed as 1 au 2 du (19.1) _ _ = v2u -7) D at az

+ I,

19.

Flat Interface

69

where

20 =-

(19.2) V is the diffusion length. For a steadily growing crystal, the diffusion field does not vary in time in the moving frame, &/at = 0, or 10

du v 2u + - -11,2= dz‘ o.

(19.3)

Taking the origin of z‘ coordinate on the flat interface, the diffusion field is solved

as u(t’) = Aexp

(-E)

(19.4)

under the far field condition ~ ( z = ’ co)= 0. The diffusion length 1~ is the thickness of the diffusion layer, and characterizes the spatial variation of the diffusion field. From the conservation boundary condition (18.7) one obtains the relation 2A V = - D ~ , , u = D- = AV.

(19.5)

1D

Thus, A = 1. From the kinetic boundary condition (18.8), the growth velocity is determined as V = K(A - 1). (19.6) Since ui = A = 1, the interface temperature in the conventional unit is Ti = T, L/C, by utilizing Eq.(18.5): The interface is heated up by the latent heat. Since Ti should be colder than the melting temperature TM for the crystal to grow, A has to be larger than 1. Even though the melt is supercooled at A > 0, the crystal with a flat interface cannot grow steadily for A < 1. The latent heat released heats up the interface so high that the heat conduction cannot transport it quick enough with small supercooling. In the case of local equilibrium ( K = oo),the time-dependent solution of the one dimensional problem is exactly obtained, and the growth velocity decreases as t-1/2.

+

Exercise: We consider the crystal growth with a flat interface under the local equilibrium condition. When the interface is moving in z direction as z = ( ( t ) , the growth velocity is shown (991 to decreases as t - 1 / 2 by assuming the scaling form for the diffusion field as , . (19.7) Show that the time-dependent diffusion equation (18.6) is transformed to the ordinary differential equation for u(w) with a new variable w = z / ( ( t ) as

d% (( du -+ -wdw2

D

dw

= 0.

(19.8)

Part IV. Diffusion-LimitedGrowth: Pattern Formation

70

(2) If u depends only on w , then show that {(t) = U(W)

dim

(19.9a)

= 2 P e P l m e-pW2dw,

(19.9b)

where P is defined by the relation

with the error function defined by (19.11) Eq.(19.9a) shows that the interface velocity decreases to zero as

0;

t-'I2.

Answer: (1) By changing variables, derivatives are related as

( 19.12) du(z/C(t)) dw du 1 du - -_ - --

d z dw

dz

a"(z/<(t)) ax2

-

1 d2u -C2 dw2 '

Cdw

(19.13) (19.14)

where ( = dC/dt. Inserting the relations (19.12-19.14) in the diffusion equation (18.6), one easily ends with (19.8). (2) By rcarrangcment, Eq.(19.8) can be written as (19.15) The first term depends only on time t and the middle term only on w , and thus should both be constant, which is set 2P. From the first equation, <(= 2DP, the interface advances in proportional to t'I2 as in (19.9). Integrating the second equality, one gets du (19.16) In - = - p w 2 + const dw or (19.17)

20. Spherical Crystal

71

With the far field condition, u ( w )= 0, it is integrated as 00

u(w) = - C /

e-Pw2dw.

(19.18)

W

From the local equilibrium condition (18.11)at the interface, z = ( ( t )or w = 1, the integral constant C is determined as

A = - C i m e-pw2dw=

(19.19)

The conservation law (18.7) is now written as

(=-DC(e-

Pw2

& a,>

=-DC-. e-" w=l

c

(19.20)

By inserting (19.15) and (19.19) in (19.20) one gets (19.10).

20

Spherical Crystal

When the crystal grows in a spherical shape, it emits heat in all directions, and it may sustain steady growth. But it will be shown that the steady growth is not possible in this case either. We study the time evolution of the radius R and its radial velocity R = d R / d t , as shown in Fig.20.1. From the symmetry the diffusion field u is expected to depend only on the radial variable r in a spherical coordinate. Then the diffusion ecluation is written as (20.1)

In the last equality the stationary approximation is used, where the relaxation of diffusion field is very quick compared to the shape variation of the crystal. The stationary distribution of the diffusion field takes the form

u ( r )= A / r

Figure 20.1: Crystal growth in spherical shape.

(20.2)

Part IV. Diffusion-Limited Growth: Pattern Formation

72

with an integration constant A. The far field condition u = 0 at r + 03 is satisfied in (20.2). The growth velocity V is determined by two ways, from Eq.(18.7) and (18.8),

as (20.3) Here the relation of the curvaturc K = 2/R with the radius of the sphere R is used. The integration constant A is then determined as A=

R 2 ( A- 2 d / R ) R+D/K

(20.4)

and the growth vclocity as

-dR= v = dt

D ( A - 2d/R) R D/K K-I

+

+R/D

(20.5)

If the crystal radius R is smaller than the critical radius pc = 2 d / A , sphere cannot grow. If R is greater than pr, thc sphere can grow. For small radius, pc << R << D I K , the kinetics rcsistivity It-' against the growth is dominant, and the growth velocity KA( 1 - p,/R) corresponds to that obtained phenorncnologically in Eq.(7.2). When the sphcrc grows largc with R >> D I K , the diffusional resistivity R I D becomes dominant and the growth rate varies as

For sufficiently large spherc as R >> pc, the growth rate V = R is inversely proportional to the radius R, and by intcgration one gets

R 2 ( t )= A D t

+ R2(0).

(20.7)

Asymptotically as t + 00, the radius incrcases in proportional to t ' I 2 , and the velocity decreases as V = R 0: t-'I2. Thus the steady growth of sphere is impossible.

21

Parabolic Crystal

In the analysis so far, ncither the planar nor the sphcrical crystals can grow steadily if the diffusion controls the growth. Is therc any shapc which allows the steady growth of a crystal? Ivantsov [89] has shown exactly that the necdlc crystal with a parabolic tip can grow steadily, if the crystal intcrface is in local equilibrium with infinitely fast kinetics and the surface tension is neglected, namely thc crystal interface is the isotherm at the melting temperature TM. If thc parabolic crystal grows steadily as

z=Vt--

x2 +

2R

r2 zi" = Vt - 2R

(21.1)

21. Paxabolic Crystal

73

Figure 21.1: Crystal growth in parabolic needle shape. The latent heat is emitted dominantly near the tip.

as shown in Fig.21.1, its height increases in proportional to the elapsed time At,

a.

whereas the width increases only in proportional to Therefore, the growth takes place mainly at the tip region, and the latent heat is emitted at the tip. The total amount of the latent heat released from the tip of parabolic dendrite down to the tail with a radius r is Lnr2V per time. The height of the dendrite is r2/2R and the surfxe area is proportional to r3/R, precisely calculated to be sr3/3R. Therefore the average heating per area is 3LVR/r, which decreases for a long parabola at r -+ 03. Heat will not accumulate and a steady growth is allowed in this shape. We now treat the problem more quantitatively. When the crystal is growing steadily in z direction with a velocity V, one transforms as usual to the moving frame of reference as (z, g, z’ = z - V t ) . For the present shape it is convenient to transform further to the parabolic coordinate (Fig.21.2)

<

= r - z’,

+

q = r z’, 6 = arctan(y/z)

(21.2)

+ +

with T = dz2 g2 z’z. (See Appendix A21). Then the diffusion equation in the steady state is described as

where 1~ = 2D/V is the diffusion length defined in (19.2). By denoting the interface as q = qi((,6,t ) , the conservation boundary condition (18.7) is written as

Part IV. Diffusion-Limited Growth: Pattern Formatioo

74

Figure 21.2: Parabolic coordinate. and the local equilibrium condition (18.11) is written as u(vi) = A,

(21.5)

since there is no capillary effect ( d = 0). The paraboloid crystal of revolution (21.1) corresponds to the interfacial shape with a constant q as qi = R. Due to the symmetry, the field u depends only on q, and the diffusion equation reduces to (21.6)

(21.7) Here the local equilibrium boundary condition (21.5) is used. From the far field condition that u(q -+ 00) = 0, the integration constant C is determined as (21.8)

From the continuity boundary condition (21.4) with qi = R and dvi/d[ = aq;/ad = aqi/& = 0, one gets the relation (21.9)

Inserting Eq.(21.7) and (21.8) into the relation (21.9)) one gets the Ivantsov relation

/Hm

A = ( R / l ~ ) e ~ " v-le-ql'Ddv ~ = P ~ P (E P )~

(21.10)

75

21. Parabolic Crystal

with the Peclet number P defined as (21.11) is the exponential integral function 111. In two dimenHere E*(P)= J,"x-'e-"dx sions, one obtains the steadily growing parabolic crystal (21.12) with the Ivantsov relation

A = GePerfc(dF)

(21.13)

with the error function defined in (19.11) [83]. (See Exercise). For a small supercooling A, Peclet number P is small and the Ivantsov relation is approximated as P ( - In P - 0.5772. . .)

A={

for 3 dimensions for 2 dimensions,

GF

and for high supercooling as A relation is approximated as

+

1, the Peclet number P is large and the Ivantsov

1 - 1/P 1 - 1/2P

A,{

(21.14)

for 3 dimensions for 2 dimensions,

(21.15)

Horvay and Cahn [83] extended the Ivantsov solution to the elliptic paraboloid: z

- Vt

1 = -(2

2R

+ 2y2) + -R2

(21.16)

with the aspect ratio (ratio of x axis to y axis) a. The undercooling A and Peclet number P = VR/2D is related as

1 A = -Pep a

L"

e-"dw

Jw[w

Pe-"

00

+ (a-2 - 1)P]= J,

J(P

+ w ) ( P + a2w)

dw.

(21.17)

For a = 1 the shape becomes the three-dimensional paraboloid of revolution and the Ivantsov relation (21.10) is reproduced. For a = 0 the system reduces to the two-dimensional problem, and Eq.(21.17) reduces to Eq.(21.13). The Ivantsov solution indicates that the steady growth is possible in the form of a parabolic dendritic crystal. But the Ivantsov relation (21.10) shows that the growth velocity V and the tip radius R are not determined uniquely for a given undercooling A. It only determines the product V R P. On the contrary, experiments show that for a given A, V and R are determined uniquely. There is some factor missing in the analysis by Ivantsov. N

Part IV. Diffusion-Limited Growth: Pattern Formation

76

Exercise: Find the Ivantsov relation (21.13) for a two-dimensional dendrite, when the surface tension is absent. Answer: Take the 2D parabolic coordinate <=l.-z', 9 = 1' z',

+

with r = d

x=fi z' = ; ( T j

(21.18)

-t)

m 5 = (< + q)/2, the diffusion equation in steady state is written as

Boundary conditions at the surface 11 = qi are expressed as the local equilibrium condition ?J(vt) = A, (21.20) and the continuity condition (21.21) By taking the parabolic shape as 17; = p , the diffusion field u depends only on 77 from the symmetry and the diffusion equation becomes simple: (21.22) The solution which satisfies the far field condition u(q equilibrium condition (21.20) at q = p is obtained as

--+ CQ)

= 0 and the local

(21.23) The continuity boundary condition, (21.21), is written as (21.24) By changing the integration variable to x = and using the Peclet number P = p/lD, Eq.(21.24) is transformed to the desired result: A = 2fie'

with the error function (19.11).

/;e-"'dx

= J?rPe'erfc(

fi)

(2125)

22. Stability of a Flat Interface

77

L

C

Figure 22.1: Isotherm distribution in the supercooled liquid around the interface (a) for a flat interface, (b) for a deformed interface without capillarity, (c) and for an interface with Gibbs-Thomson effect included.

22

Stability of a Flat Interface

In the section 19 we discussed the steady growth of a flat interface and found that for A < 1, no steady state is possible. If the interface remains to be flat, the growth will slow down, and the growth velocity becomes zero even with a finite driving force. Another possible scenario of the time evolution of the flat interface is that it looses stability and deforms. We now study the latter possibility. In order to discuss the extreme case of the diffusion limited growth, we assume hereafter an infinitely fast kinetics at the interface, K + 00, and the local equilibrium condition (18.11), or (22.1) U;= A - d K . First we give a qualitative explanation on the stability of the interface. If the flat interface moves steadily with a velocity V , the diffusion field u varies from A to 0 in a length scale of diffusion length I D = 2D/V. The isotherm u = const lies parallel to the interface z' = z - V t = 0 as schematically shown in Fig.22.la. When a part of the interface advances faster than the other by some fluctuation, what will happen to this pointed part? First we neglect the small effect of surface tension or capillarity. Then at the interface the diffusion field u;takes a constant value A, irrespective of its deformation. Near the pointed part the isotherm ui = A pushes forward other isotherms, and the isotherm density becomes high here, as shown in Fig.22.lb. The high density of isotherm means the steep slope of Vu, and the heat flux -DVu increases and releases the latent heat quickly. The pointed part can grow faster than the undeformed part, and the deformation is enhanced: The flat interface becomes unstable. This instability is first studied analytically by Mullins and Sekerka, and is called Mullins-Sekerka instability [141, 1221. So far we considered the interface instability when the crystal is growing in the supercooled melt and the heat is transported through the liquid. If the crystal is supercooled and the melt is hot, what will happen for the interface deformation?

Part IV. Diffusion-Limited Growth: Pattern Formation

78

C

L

C

Figure 22.2: Isotherm distribution when the crystal is supercooled. (a) A flat interface, and (b) a deformed interface without capillarity. Since the crystal is supercooled, the latent heat is released through the crystal. For a flat interface isotherms in the crystal are running parallel to the interface, as shown in Fig.22.2a. If a part of the crystal grows fast at some point by fluctuation, the separation between isotherms near the pointed part increases and the isotherm density decreases (Fig.22.2b). The slope of the isotherm decreases, and the latent heat is less effectively transported. Thus the growth rate there decreases, and the interface recovers the flat profile. In this case the interface is stable against fluctuation. There is an asymmetry and the instability occurs when the interface is propagating in the region where the transport is taking place. We now consider the effect of surface tension on the stability of a flat interface. The surface tension lowers the degree of supercooling at the pointed part with positive curvature ( K > 0) as (22.1) and shown in Fig.22.1~. Since the driving force at the pointed part decreases, the growth rate there decreases. Thus the surface tension acts as a stabilizing factor for the flat interface. The total stability is determined by the competition between the diffusional destabilization and the energetic stabilization. We now describe the stability in the linear analysis. The interface is assumed to be deformed sinusoidally with a wavelength X = 27r/q as ((s, y; t ) = V t

+ a, exp(w,t) cos(qs).

(22.2)

The amplification rate wq determines the stability of the flat interface: When wq is negative, the interface is stable, whereas when w, is positive, it is unstable. Since the interface deformation influences the diffusion field u, it also has an additional variation to that of the flat interface in Eq.(19.4) as U(X,y, Z ; t ) = A CXP

+ Gu,exp(w,t) cos(qs) exp [-A,(t- - V t ) ]. (22.3)

Here A, should be positive, since the far field u(z + 00) will not be affected by the interface deformation and the deviation should decay in z direction. By inserting

22. Stability of a Flat Interface

79

Figure 22.3: Dispersion relation of a sinusoidal deformation of a flat interface. (22.3) in the diffusion equation (18.6) and by considering the terms containing cos qx, one gets the relation w, 212, - - -q2 + A:. (22.4) D

+

1D

From the local equilibrium condition (22.1) with the curvature K M --d2z/dx2, A should be equal to A and the deformation amplitude Su, and a, are related as (22.5) In the linear analysis the anisotropy in surface energy y gives no contribution and do denotes the isotropic part of the capillary length defined in (18.10). From the energy conservation or continuity equation (18.7), one gets the relation that A = 1 and

3 =D

(2)2+ (t

-do$-) A,.

(22.6)

From Eq.(22.4) and (22.6) the damping in z direction is determined as (22.7) and the amplification rate as

which is depicted in Fig.22.3. For slow growth, the diffusion length ZD = 2 D / V is large; l ~ / d o>> 1. For a deformation with a wavelength less than ZD or q l D > 1, the dispersion relation (22.8) is approximated as (22.9)

Part IV. Diffusion-Limited Growth: Pattern Formation

80

For wavenumbers q larger than the stability value (22.10) the deformation damps down by the negative amplification rate w,. For the small wavenumbers q, however, the capillarity is not strong enough to stabilize the flat interface, and w, is positive. In terms of the wavelength, the mode with a modulation wavelength longer than the stability length (22.11) is amplified. The most unstable mode has the wavenumber (22.12)

Qnt

= qs/&

urn

= ZVQm

with the amplification rate

2

(22.13)

as is shown in Fig.22.3. So far we considered the one-sided model. If the diffusion constant in the crystal is the same with that in the liquid phase, namely in the symmetric model, thc stability length is given by A, = 2 a m . Exercise: Crystal in a spherical shape is growing in a static diffusion field V2u, = 0 with the local equilibrium boundary condition (18.11). Discuss its stability against the deformation as (22.14) ~ ( 04)%= R(f) aim(t)Yrm(@, #) with a spherical harmonics in spherical coordinates ( ~ , t l#) , [140]. The Laplaciui in spherical coordinate is expressed as

+

xm

(22.15) and spherical harmonics k'jm are eigenfunctions of the angular part value -t(Z 1) as Ak'jm(~,@)= -111 1)Ylm.

+

with an eigen-

+

Also in the linear approximation the curvature K ( @ ,4)can be represented as

(22.16)

22. Stability of a Flat Interface

81

Answer: The diffusion field is modified from Eq420.2) as A

u(r,e,4) = 7 + h m ( r , t ) K r n ( e , d ) .

(22.18)

Since the field u(r,8,4)should satisfy the diffusion equation in the static approximation V2u = O or (22.19) - E(Z 1)fiulm = 0,

( ~ 2+ )

the deformation amplitude which decays far from the crystal is solved as

B

6Utm

= ~l+l'

(22.20)

The local equilibrium boundary condition (18.11) at the interface (22.14) is explicitly written as 1 arm B u,. - A ( -R_ z x m ) R'+2Ejm = A - d 2)(1- l ) a l m ~ m ] (22.21)

+

in the linear approximation, and two parameters A and B are determined as

A=R(A-:)

B = RI-' [ A- ( I

(22.22)

+ 2)(1 - l)dl aim = R'

(22.23)

The continuity equation( 18.7) is expressed as

v, = R + hlmKm.

(22.24)

From the zeroth order, one gets the growth velocity (22.25) From the linear order in atrn, the modulation velocity is obtained as

For small radius R when wtrn < 0, the deformation amplitude arm decays and the spherical shape is stable. For large radius when ulm> 0, the deformation grows and the spherical shape becomes unstable. The critical radius for the 1-th mode is given from the relation wlm = 0 as

&(E)

d A

= -(Z2

+ 31 + 4) =

pc

l 2 + 31

+4.

(22.27)

Since the 1 = 1 model is marginal as ulm= 0 , the 1 = 2 mode is destabilized first at the radius R seven times of the critical nucleation radius pc = 2d/A. Thus a small sphere loses stability in the shape, and starts to deform.

Part IV. Diffusion-Limited Growth: Pattern Formation

a2

Figure 23.1: Fractal dendrite of Au deposited on Ru [86].

23

Fractal Dendrite

If there is no surface tension

($0

= 0), the dispersion relation (22.8) reduces to

%- 2 - - -1411 D lD

(23.1)

which is positive for all deformation wavenumbers q. The interface is always unstable, and it is most unstable for the largest (qI or the finest structure of deformation. From Eq.(22.27) the spherical crystal is also unstable for any size if there is no capillarity effect. Therefore, one expects that the very fine and irregular structure will be realized for the crystal growing in the diffusion field without the surface tension. This kind of structure is actually observed in the vapor deposition on a very cold substrate, as shown in Fig.l.3a [41] and Fig.23.1 [86]. The adsorbed atom can diffuse randomly on the substrate, and its life time is very long on a cold substrate. In the meanwhile, the adatom collide with another adatom, coagulates and stop moving. Since the substrate is so cold that the once coagulated atoms never dissociate again to search for energetically more favorable position, the surface energy cannot play its role to stabilize the interface morphology. The irregular structure so grown is self-similar and called fractal, and is intensively studied recently (189, lo]. There are many other examples of fractal objects as aggregates grown by electrochemical deposition [130, 211 and bacteria colonies [132, 221. We summarize the fractal theory relevant to crystal growth, and discuss fractal-to-compact crossover for the aggregate growing in a diffusion field of a finite density [191, 1851.

23.1

Diffusion-limited Aggregation (DLA)

The irregular aggregate is first studied theoretically in thc computer [197). An immobile crystal particle is placcd at some point as a seed, and a randomly diffusing gas particle is released far from the seed. When the gas particle comes into contact with the seed particle, it freezes and is incorporated into the aggregate. Then, the

23. fractal Dendrite

83

Figure 23.2: Diffusion-limited aggregation (DLA) grown in computer [150J. There are (a) 6 x lo4, (b) 6 x lo5 and (c) 6 x lo6 particles in each cluster. next gas particle is released far from the aggregate, and performs the random walk. After it makes contact with the aggregate, it freezes, and a new particle is released again. By iterating this procedure, the aggregate grows by incorporating atoms one by one. The grown aggregate is called the diffusion-limited aggregate (DLA), and its structure is shown in Fig.23.2. It is irregular, ramified and very open. The part looks similar to the whole by enlargement, and there is no characteristic length in the structure except the lower and upper cutoff lengths, the atomic and the system sizes. This self-similar object is called fractal. If there are N ( r ) atoms in a region with a radius T , the number of atoms in the radius br is N(br) = b D f N ( r ) .By taking b = T - ' , then N ( r ) = rDfN(1) (23.2) and Df is called the fractal dimension. If the object fills the d dimensional space homogeneously with a finite density, Df = d and it is called compact. DLA is, on the contrary, fractal with Df = 1.71 in d = 2 dimensions, and with Df = 2.49 in d = 3 dimensions [134]. Since there are N rDf number of particles in a radius T , the aggregate density is N n(r) +d, (23.3) N

N

N

Td

which becomes zero asymptotically for T + 00. It means that the aggregate has many open spaces. Zero asymptotic density is quite imaginable because the aggregate grows from a diffusion field of zero density: There is only a single gas particle during the whole growth simulation. Also a particle can walk forever until it comeg to contact to the aggregate, and it means that the aggregate grows very slowly, V + 0. The diffusion equation thus reduces to the Laplace equation V2u = 0. The diffusion length I D = 2D/V is infinity, and the surrounding field as well as the aggregate have no characteristic length in the DLA growth.

Part 1%’.Diffusion-Limited Growth: Pattern Formation

84

0

Fractal

fD Compact r

Figure 23.3: Schematic behavior of an aggregation density n(r) growing in a diffusion field of a finite gas density ns. The diffusion length ID is the characteristic length scale of the crossover between the fractal and compact structure.

23.2

Fractal-to-Compact Crossover of an Aggregate in a Finite Gas Density

In a real experiment an aggregate grows from a gas with a finite density n,, whose time evolution follows the diffusion equation. The growth rate of the aggregate V should no vanish, and the relation between the velocity V and the gas density ng is of interest [185]. To realize the steady growth easily, we consider a unidirectional growth of an aggregate from a linear seed, similar to the “fractal forest” shown in Fig.l.3b. In front of the crystal, there are gas particles with an average density ns and they are making random walks on a square lattice to the nearest neighbor site. When a gas particle comes into contact to a seed or to an aggregate grown from it, it freezes and becomes a member of the aggregate. The time scale is so chosen that in a unit time every gas particle makes one diffusional jump on average. By taking the lattice parameter as a unit of length, the diffusion constant D is 4. The aggregate front h(t) is defined as the height of newly crystallized particle, and the velocity V is obtained from the time variation of h(t). Since the gas particle is incorporated in the aggregate instantaneously and irreversibly, the density in front of the aggregate vanishes. It relaxes back to the given density n, within the distance of diffusion length ED = 2 D / V . Within this length, the gas density is low and the situation boks similar to the DLA growth: A gas particle sticks to the aggregate randomly and never melt back. Therefore, the grown aggregate has a fractal structure. But, since the aggregate grows from a gas with a finite density ng,the material conservation does not allow the aggregate density to diminish as n ( r ) rDc-d like the fractal object in the previous subsection does. For large distance r -+ 00, the aggregation density should saturate to n,, as shown in Fig.23.3. A crossover of the aggregation density from the power , the law decay (23.3) to saturation takes place within a characteristic length 1 ~and

-

23. Fractal Dendrite

85

Figure 23.4: Irreversible and unidirectional solidification of aggregate from a gas with finite density: (a) ng = 0.08, and (b) ng = 0.1 [185].

1 V

0.I 0.01

0.I

0.5

I

ng

Figure 23.5: Velocity of the aggregate front u versus the gas density n,, showing the power law dependence v -, n; with Y = I/(d - Df)in d = 2 dimensions with Df= 1.71 [lSS]. relation

l - ( d - Dr )

D

729

(23.4)

is expected. There should be a crossover from the frwtal to compact structure at the length scale of order lo. This gives the relation between the aggregate growth velocity V and the gas density ng as

In two dimensional system (d = 2), Monte Car10 simulation shows that the aggregate consists of many branches of irregular dendrites, and the separation between branches decreases as the gas density ng increases, as shown in Fig.23.4. This is compatible with the decrease of the characteristic length as ng increases. The growth velocity of the

Part IV. Diffusion-Limited Growth: Pattern Formation

86

aggregate V is found to increase in powers of the gas density ng with the exponent compatible with the fractal dimension Dr = 1.71 in two dimensions, as shown in Fig.23.5. The scaling (23.5) is recently confirmed in the deposition experiment of silver metal leaves from AgNOB solution [139].

24

Capillary Effect and Regular Dendrite

Without the surface tension, the crystal profile is shown to take a random and irregular form. In reality, most metals and some plastic crystals growing in a free and open space takes a regular dendritic form with a stable tip oriented in a special crystallographic direction, as shown in Fig.24.1 [84]. The tip of the dendrite is parabolic and the tip radius R and the growth velocity V satisfies the Ivantsov relation (21.10) with the undercooling A (Fig.24.2) [84]. In experiments the undercooling A determines R and V uniquely, contrary to the Ivantsov solution where an infinite degeneracy is expected. Capillarity may select a unique operating point among multitude of possibilities. It introduces a new characteristic length, the capillary length d = ;j.C,,TM/L2 of (18.10). It is intrinsic, since it is determined only by the material parameters and every other length as p or ED is expected to be scaled with d. Since the main interest lies in the interface structure, it seems advantageous to integrate out the diffusion field and to derive the interface dynamics. Many local models for the evolution of the interface profile were proposed, and gave contributions to understand the problem of the velocity selection [40, 17, 18, 100, 1011. But the true interface dynamics is nonlocal even in quasi-stationary approximation, as explained in detail in Appendix A24.1. We summarize the results of recent studies on the nonlocal model.

24.1

Velocity Selection by Microscopic Solvability

Due to mathematical simplicity, the analysis is mainly done on two dimensional dendrite [124, 105, 33, 1561. The capillary length is defined in terms of the twodimensional surface stiffness The surface tension p is assumed to have four-fold rotational symmetry reflecting the crystalline order of, such as, succinonitrile.

a.

p(6) = po(l 4-zcoS48).

(24.1)

For positive Z, B = 0, f ~ / 2 ,and x corresponds to the maximum of the surface tension, and to the corners in equilibrium shape (Fig.6.5). The surface stiffness is expressed as

p =p

+a2p = po(l - 15Sc0848), a82

(24.2)

and the capillary length (18.10) is then written as

d

do(1- fCoS48)

(24.3)

24. CapiUary Effect and Regular Dendrite

Figure 24.1: Dendrite of succinonitrile [84]. I t

I IIIII 10-1

10-4

10J

I

,‘

I

I , , L 11

TIP RADIUS (cml

Figure 24.2: Velocity V versus the tip radius R for a given undercooling A [84].

Part IV. Diffirsion-Limited Growth: Pattern Formation

a8

with do being the magnitude of the capillarity lciigtli and E = 15C thc strength of the anisotropy. Oricntatioiis 0 = 0, h / 2 , 7 r correspond to the rninimum of the stiffness and the capillary length. By including thc capillarity, tlic stability length appears, for example, A, = 2 7 r , / m in the one-sided modcl. When the tip radius p is smaller than A,, the tip is unable to grow further due to the capillary cffcct, ;wid the interface turns to be flat. When p is larger than A,, thc tip is unstable to thc deforination due to the Mullins-Sckcrka instability. Thcrcforc, the dendrite tip radius is expcctcd to be of order A,. Then, oiic introduce a dimensionless paramctcr (24.4) which is called the stability parameter. Since the capillary length do is small, this CJ is small, and one may try thc expansion of the profile in powers of C J .However, as the capillary cffcct is contained in the Gibhs-Thomson condition, it is coupled to thc curvaturc K , the highcst derivative of the interface profile as O K -aa2z/dx2. In this casc, nornial perturbation fails and one htls to use a singular pcrturbation [16]. Thcre are inany theoretical works on the dendrite theory which x c sunimarized in many reviews [124, 105, 33, 1561. In solving tlic nonlocal interface equation, boundary coiitlitions has to be considered appropriately. Far down t h e dendrite tail, thc curvaturc is so small and tlic profilc should approach to thc Ivantsov parabola. By iiitcyy-ating the intcrfacc profile from thc tail to the tip, one gets in gcneral a finitc slope (3 at the tip iL1ld tlic solution cannot hc cxtcndcd syiriirictriciilly to the other sidc of tlic tail. Slopc at, t,hc tip can vanish only for spccial choicc of tlic growth velocity u or thc stability ptirametcr C J . This is called the solvahility condition [124, 105, 33, 15G], autl is skrtchcd in a littlc inore detail in Appendix A24.2. Hcrc! I suniinarixc tlic niain conclusioiis.

-

1. Without surface tinisotropy growth.

= 0, there is no strxly state for tlic dendritic

2. With an anisotropy E # 0, thc syiiiinctric dcndritc1 grows s t d i l y with its tip oriciitctl i n the dircction of the stiffness minimuni 8 = 0.

3. The stability paranictcr CT is found to dcpcntl only 011 iiii anisotropy parameter t , but is indcpciidcnt of the niidcrcooling A. For sinall E , CJ is iqqxoximately and thus ,upz rcmains constant for various iiridcrcooling A: proportional to f7I4 (24.5) The value of ~ ( t for ) tlic out-sided iiiotlcl is shown lo be twice as large as that of tlir syinmctric modcl [138]. By coinbilling with thc Ivantsov relation, v p = 2 0 P ( A ) , the growth rate is obtained as 71 M P 2 . For d = 2 it reads as v M A" at sinall undercooling where the approximation (21.14) holds.

24. Capillary Effect and Regular Dendrite

89

b)

a)

Figure 24.3: Simulation of the dynamical evolution of the dendritic crystal at (a) = 0 (isotropic surface energy), and E = 0.10. The isotropic surface energy (a) shows the tip splitting, and the anisotropic one grows stably in (b) [165]. E

,

0.08

1:' /

1.=?* A 0

~0.25

0.02-

-00

0.05 '

-

I

010

035

Figure 24.4: The stability parameter c7 dependent on the anisotropy parameter two different undercoolings A = 0.25 and A = 0.50 [165].

E

at

This result is confirmed by the numerical simulation by solving the diffusion equation in a quasi-stationary approximation by the boundary element method and integrating the shape evolution [165], as explained in Appendix A24.3: For E = 0, the dendrite tip splits as it develops in time (Fig.24.3a), and for E > 0 the tip stable dendrite grows as shown in Fig.24.3b. The parameter u = dov/2DP2 is found to depend on anisotropy E but is independent of the supercooling A, as shown in Fig.24.4. In the three-dimensional experiment of succinonitrile dendrite growing in its melt

Part IV. Diffusion-Limited Growth: Pattern Formation

90

SUPERCOOLING (C)

am

,

.,

, . t

0.01

,

* , * 0.1

NORMALIZED SUPERCOOLING

Figure 24.5: The stability parameter P41.

at various undercooling A's for succinanitrilc

(Fig.24.1) (841, the relation V R 2 = const is found to be satisfied as shown in Fig.24.2 and Fig.24.5. The strong anisotropy dependence of VR2 is, however, not observed experimentally. There is somc argument that anisotropy is not necessary in the mode selection [71], but at least it seems plausible that the dendrite grows in thc direction of thc stiffness minimum, since the minimum stiffness means less effective in suppressing deformation and the deformation grows fastest in that direction. If therc is no prefcrence in orientation, the growing tip is very susceptible to the orientation fluctuation, and may split or form irregular dendrite. The chaotic behavior of the isotropic interface is already found in the KuramotoSivashinsky equation discussed in section 17. There is another example of tip instability in an isotropic system, a viscous finger problem [162, 1521: A viscous fluid is confined in a narrow gap between two glass plates of a Hele-Shaw cell [75]. When the fluid with low viscosity (like air) is pushed into the fluid with high viscosity (like water), the meniscus between two fluids is unstable and forms finger-like pattern, known as the viscous finger. The problem is formulated similar to the dendritic growth. Onc essential difference between the two problems is that the surface tension is isotropic in the viscous finger problem, whereas it is anisotropic in the crystal growth. The pattern realized in the radial Hele-Shaw cell shows a branched ramified structure as shown in Fig.24.6 [152]. By engraving a line groove in some portion on one. glass plate of a Hele-Shaw cell, the tip stability is realized as shown in Fig.24.7a [131]. The fluid dendrite thus obtained satisfies the relation vp2 = constant as shown in Fig.24.7b, until the tip radius p becomes too small. This experiment clearly shows the importance of the anisotropy in stabilizing the dendrite tip 1191. So far, theories and simulations dealt with two-dimensional(2D) dendrite growth. Recently, solvability condition is successfully applicd to the three-dimensional (3D) dendrite [14, 361. The surface frec energy $6, #) now has an orientation dependence on two Euler angles, B and $C of the normal vector to the surface. In the theory, a cubic anisotropy -y(O,$) = 1

+ C(4 em48 + 3 sin4B + sin" 6 cos 44)

(24.6)

24. CapiUiuy Effect and Regular Dendrite

91

Figure 24.6: Viscous finger pattern in a radial Hele-Shaw cell [152].

0001

0.01

01

1

10

100

T I P SPEED V ( c m / s )

Figure 24.7 (a) Viscous finger with a groove linc cut on one glass plate. Along the groove grows a tip-stable parabolic dendrite, where&? other tips are unstable to splitting. (b) Log-log plot of the tip radius of the fluid dendrite versus tip speed [131].

Part IV. Diffusion-Limited Growth: Pattern Formation

92

is assumed. The selection problem is solved in two stages: Near the dendrite tip, the shape is perturbed as T2

+

A4r4 cos4#+ .. . (24.7) ZR with r2 = x 2 + y2. The growth rate V and the tip radius R is found to satisfy the same relations as in the 2D case: the Ivantsov relation A V R = 2 D P ( A ) N -(24.8) Inn’ for small supercooling A, and the solvability scaling relation t = --

(24.9)

Therefore, the anisotropy dependence of the velocity V and the tip radius R is expected to be the same with the 2D case 1141. The analysis near the tip cannot be extended to the dendrite tail straightforwardly, since the correction A4r4 cos 4# grows faster than the unperturbed Ivantsov paraboloid -r2/2R [36]. But the correction form indicates that four fins extend in the directions, q5 = 0, ~ / 2 T, , 3 ~ / 2 .When one looks far down the dendrite, the crystal interface is almost parallel to z axis, and the diffusion field hardly varies in z direction; a2u/i3z2M 0. Then the evolution of the dendrite fin in the tail region is essentially controlled by the two dimensional diffusion [36]. The four fins that have started to grow near the tip develop into two-dimensional parabolic dendrites in sy cross section. For example, down at a height z(< 0) from the tip, a dendrite growing in x direction takes a form (24.10)

where p and v are the selected values of radius and velocity of the 2D dendrite. Here the time t is replaced by the height It1 divided by the 3D growth velocity V, since the profile at a height z develops to that at a height z - Vdt after a time d t in the steady state Between the two extreme asymptotics, Eqs. (24.7) and (24.10), an intermediate asymptotics is found [36] until the arm length exceeds the 2D diffusion length. In this intermediate region four fins affect their growth mutually. At a height 1x1 from the tip, the total cross-sectional area of fins should be that of the Ivantsov paraboloid for the steady growth. If the fin protrudes a distance s with a width y, then the area is about sy 12). Even though x and y depends on the time 121, the dimensionless parameter of the system

-

(24.1 1)

-

-

may remain constant [5]. By assuming the profile 5 lzlu and y a2 is calculated as 02 Jz13-5a. If u2 remains constant, then a = 3/5, and the arm length N

24. Capiitary Effect and Regular Dendn'te

93

-

Figure 24.8: (a) Xenon dendrite and (b) tip contour compared with a power law fit to parabola and I x1.67[261.

-

x grows as x )2)3/5 and the arm width y increases as y 1z12/5. This behavior is originally found in the two-dimensional Hele-Shaw flow problem numerically and analytically [5], and later applied to the dendritic growth [36]. The profile of the fin in xz cross-section is written as z x5/3 in this intermediate region. The profile is recently observed in the experiment of Xenon dendrite as shown in Fig.24.8 [as]. When the arm length exceeds the 2D diffusion length, the final aymptotics sets in and the arm length grows with a constant velocity. For the details, see thc references 114, 361. N

N

Tip Stability and Sidebranches

24.2

With an anisotropy E in the surface tension, the parabolic crystal with a tip radius R is found to grow steadily with a velocity V in the supercooled melt with the supercooling A. V and R are uniquely determined from both the Ivantsav relation V R = 2DP(A) and the solvability condition VR2 = 2Ddoo-'(~). However, the dendrite is growing in the environment with fluctuations, for instance, created by thermal noise or by hydrodynamic convections. Is the tip stable against these fluctuations? Another problem is the origin of the sidebranches. According to the solvability theory, the tip of the selected dendrite is almost parabolic without sidebranches. How, thcn, are the sidebranches created in experiments? If the crystallizing front is flat, Mullis-Sekerka instability takes place and the front is unstable. It is most unstable against the fluctuation with the wavenumber qm = with the exponential amplification of the amplitude with the rate w,, = 2vqm/3, as explained in the section 22. But actually the dendrite tip is curved parabolic as

,/-

'X

z = C(z) = vt - 2P

(24.12)

Part IV. Diffusion-Limited Growth: Pattern Formation

94

Figure 24.9: Trajectory z = E(z) of the dcformation on the interface, which is alway perpendicular to the interface profile, 2 = ((x). in two dimcnsions. Sincc the interface deformation grows in thc normal direction n = (n,, n,) cx (-d(/dx, 1) as shown in Fig.24.9, the node of deformation follows the trajectory 2 = ((2)

(24.13) or, by integration, [(x)=plnx+C.

(24.14)

When the initial deformation at t = 0 is given at position (xo, - s i / 2 p ) on the intcrface, the integration constant C is determined as C = -x:/2p - plnxo. At time t , the deformation has bcen propagated at the site (z, ((x)) determined by crossing point of C(x) and [(T) as (24.15)

m,

Asymptotically for large t , x increases as but its height 2 = ((2:) increases slowly as p/2.1n t . The vcrtical scparation between thc dendrite tip at (0, vt) and the deformed position at M p/21nt) increases as u t . Though t h t deformation amplifies with the rate w,, it is convected down with a velocity v and the tip remains stable. This is called the convective stability of the dendrite tip. Down convected noise is amplified to form sidebranches. If this scenario is correct and sidebranches are originated from the random noisc at the tip, sidebranchcs on the different side of the dendrite arc expected to be uncorrelated. By measuring the correlation the noise origin of sidebranches is confirmed [54]. The external oscillatory flow imposes systematic fluctuation at the dcndritc tip, and thc synchronized formation of aidebranches is observed 1281. Of coursc, the tip stability depends on thc amplitude of the initial noisc. If the initial noisc is too strong, the cleformation is strongly amplified.

(m,

24. Capillary Effect and Regular Dendrite

95

Figure 24.10: The sidebranch periodicity X near the dendrite tip is proportional to the wavelength of the most unstable mode A, p f i , irrespective of the undercooling A and the anisotropy E [165]. N

When the deformation amplitude becomes larger than the tip radius p before the deformation is convected down the length of order p, the dendrite tip splits [37]. In fact, the fractal structure is realized for NH4Cl dendrite growing in a Hele-Shaw cell when a bottom glass is cut rough randomly [82). We now study the sidebranch formation more quantitatively. The convected deformation amplifies and becomes a sidebranch with a periodicity X about that of the most unstable mode, (24.16)

In the numerical simulation, the wavelength X of the sidebranch is defined by dividing the growth rate v by the number of sidebranches produced per unit time. The ratio of the sidebranch periodicity X to p f i was fount to be independent of the supercooling A and the anisotropy E as shown in Fig.24.10 [165]. As the perturbation slides down along the side of the dendrite z = -x2/2p, the normal velocity vn decreases as v, = -v(a(/&)-l as shown in Fig.21.1. N

vm,

Then the local diffusion length increases as ZD(~) = 2D/v,(z) the most unstable wavelength

N

Z

D as well~ as

(24.17) The periodicity is expected to increase as the sidebranching is convected down the shaft of the dendrite. Along with the variation of thc periodicity Xm(z), the amplification rate w,(z) = 2vn(z)qm(z)/3 of the most unstable mode varies while the

96

Part IV. Diffiwion-Limited Growth: Patteni Formation

1

0.5

E" r'

0

v

m

-0.5

-I - 1

. I .5 0.75

1

5:14

1.25

1.5

Figure 24.11: (a-11) The time scqiiciicc of tlic disturbance arnplificatiou iicu the tip of the dcntlritc to the sidelmuiclch. (i) Thc inaximurn curvatiirc f i t vcrsus the arc length s [157]. K I is proportional to the noisc amplification. perturbation is corivcctcd down. (24.18)

Thcrcforc, when the perturbation is coiivrctcd down to a hcight Iz.1 during a tirric t = 121/t~,the deformation is totally amplified by a factor A ;

Precise expressions of A arc obtaincd for the two-dimensional dciidritcs [ 155, 111, and for the three-dimensional axisyinirictric dendrite [123]. If the initial perturbation is thc thermal noisc at the dendrite tip, it is found to be too small to explain the observed

24. Capillary Effect and Regular Dendrite

97

position of the A t s t sidebranch. On applying a noise near the tip by a localized heat pulse, the noise amplification to the sidebranch (24.19) is studied experimentally [157]. The maximum of the curvature on the sidebranch is found to increase as ic1 exp(as'i4) during the noise is convected down the dendrite till the arc length s, as shown in Fig.24.11. As explained in the previous subsection, there is the latest theory of the threedimensional non-axisymmetric dendrite with four fins formed a little down the tip (361. Since the profile of the fin t = [(s) 25/3has a slope as a [ / a x N t2l5which is milder than that of the parabolic dendrite d(,v/dx .z1/2, the convective effect is small and the dendrite is more susceptible to thc noise. At the height ( z ( ,the most unstable mode has the short wavelength X,(z) zl/', the large amplification rate u m ( t ) 2-3/5,and the large total amplification factor A exp [ C O - ' / ' ~ ~The /~]. more precise formula is given by Brener and Temkin [38]. The amplification factor A increases steeply along the fin compared to the parabolic case, (24.19). Thc first side branch is in fact observed on the fin of the Xenon dendrite at the position expected from the thermal initial noise [26].

-

-

-

-

24.3

-

-

Dendrite in a Channel

For the case of a free dendrite, the anisotropy of the surface energy is shown t o stabilize the dendrite tip. When the crystal grows in a channel with non permeable walls, interaction with the channel walls through the diffusion field allows stable stationary patterns even for a system without anisotropy [104]. The wall provides the effective anisotropy in the system and the steady growth of a symmetric finger-shaped crystal is possible if the dimensionless supercooling is large, A > [31]. Due to the global conservation of the energy, the width w of a symmetric finger in a channel of width X should be equal to AX; w = AX. If A is small, thc symmctric finger is far from the wall and it cannot maintain the stable form. Therefore, A should be larger than 1/2. For a fixed supercooling A, thc growth velocity v varies as a function of the channel width X as shown in Fig.24.12. The rcsult is obtained by 0.03,

*

1

* 7--r

1

-I---

0

-

.-,0.02

-8 ?

-

1 -

0.01 -

0

-

'

j

'

"

'

'

Pa& IV. Diffusion-Limited Growth: Pattern Formation

98

Figure 24.13: Asymmetric finger along the wall [35]. the extension of the numerical simulation described in Appendix A24.3. Velocity v has a maximum, and it decreases for a large channel width A. For an infinitely wide system, as we discussed section 24.1, the isotropic dendrite cannot grow steadily with a finite velocity. When the width becomes too wide, the finger splits to have shorter period. Solvability theory by Brener, Geilikman and Tcmkin [31]found the maximum It is natural velocity v ( D / d o ) ( A- $)’f2 at the channel width X &(A t o assume that the state with a maximum velocity is selected and realized. In the simulation, however, the asymmetric dendrite is observed which grows along the wall, as shown in Fig.24.13 1351. In the simulation a mirror boundary condition is used: u(x, y) = u(--z, 9 ) = zl(X - 2,y) for the diffusion field and ((z,yy) = <(-z,y) = <(A - z,y) for the interfacial profile with 0 < x < X/2. This asymmetric dendrite along the wdl thus really means a double-finger structure with its mirror image at negative x. This structure is called “doublon” for short [87]. In a time-dependent simulation model [87], the doublon is also found in the system with periodic boundary conditions. If two fingers are growing side by sidc, one generally anticipate that the one which steps little ahead by fluctuation wins the competition against the other by gaining more diffusion field supply. In the present situation, when one wins, the periodicity X increases and the growth rate v decreases according to Fig.24.12. The one ahead is caught up by the nearby finger. Thus the doublon can survive as a stable profile even at A < $. Recent solvability analysis shows that the doublon grows with ~ } Aanganisotropy [15]. the velocity v proportional to v ( ~ ~ ~ without

-

N

3)r5f2.

-

24.4

Morphology Diagram

From the findings in the previous subsections, an isotropic crystal (if it exists) can grow radially outward with a doublon structure in an open space. Since the separation

24. Capillary Effect and Regular Dendrite

99

Figure 24.14: (a) Equilibrium shape of a crystal coexisting with lattice gas atoms in a closed system. Simulation results (symbols) for different Ap are compared with thc theoretical results drawn by curves. Time evolution of a crystal in an open system are depicted at different driving forces; (b) dendritic shape at Ap/lc~T= 10, and (c) irregular structure with tip splittings at Ap/k,T = 12 [166]. between two double-fingers widens as they grow, the space should be filled by new fingers developed from sidebranches. The structure as a whole should have a spherical and convex envelope, and is called the compact seaweed (CS) structure [34] or dense branched morphology(DBM) [ZO]. With an anisotropy, dendrites grow in directions of stiffness minimum. When the dendrite tips are separated radially, the secondary arms from the sidebranches fill the space with a concave envclope. This is called the compact dendrite (CD) (341. The morphological transition from CD to CS pattern is first obtained in the Monte Car10 simulation of acrystal growth from a lattice gas [166], as is shown in Fig.24.14. The growth is similar to the DLA growth from a finite density gas, but with an

100

Part IV. Diffusion-Limited Growth: Pattern Formation

Figure 24.15: Morphological transition from (a) dcndritc, (b) intcrmcdiate, and (c) dense branchcd morphology (DBM) of a hcxatic liquid crystal [151]. Undercoolings are (a) A = 0.24, (b) 0.45 and (c) 053. interfacial cncrgy includcd. Thc interaction is the sanic with the Ising model (6.33) such that a brokcn bond from a crystal atom costs an cnergy J . When the gas atom diffuscs and touches thc crystal interface, it tries crystallization. If it crystallizes, thcrc is a chemical potcntial gain -Ap but has to pay an cnergy cost (2/2 - n)J if thc crystallized sitc has n ncarcst ncighbor crystal atoms m o n g z coordination number. Thesc energy changcs should be taken into account in the Boltzmann weight as is described in the subsection 9.1. One also has to considcr thc mclting process from the crystal surfacc to satisfy the dctailcd balancc. In a closcd system whcrc the total numbcr of gas and crystal atoms is fixed, an cquilibrium shapc is realizcd as is shown in Fig.24.14a. In an open system, the gas atom is fed from a particle rcscrvoir with a fixed dcnsity at a distance far from the growing crystal. Then thc crystal grows steadily. For small A p as n p / k ~ T= 10, thc crystal grows in diagonal [ l l ] dircction in a rcgular drndritic form, as shown in Fig.24.14b. At a larger A p as A p / k B T = 12, the crystal grows in an irregular form with concave cnvclopc, as shown in Fig.24.14~.Thus simulation clearly shows a crossovcr in the growth morphology. Therc arc niany similar simulation works on the morphological transitions [ 175, 1761. Thcrc is also an cxpcrimcntal observation of thc morphological crossovcr in the growth of columnar hexagonal crystal, as shown in Fig.24.15 [151]. Analytical studics arc also pcrformcd on thc dynamicd selection of morphology [34]. It is natural to assume a maximum velocity critcrion that the pattern with maximum vclocity is sclcctcd. With this hypothesis, Brencr, Tcmkin and MiillcrKrumbhaar have dcrivcd the morphological phasc diagram in the pliasc space of the supercooling A and thc anisotropy E , as shown in Fig.24.16 [34]. Since they didn’t considcrcd doublons in an isotropic or weakly anisotropic region, CS structure can appear only for A > 1/2. By coiisiclcring the double-finger structure, the crossover between thc CS and CD structurc is cxpccted to take place at A c7/20 . At small undcrcooling A and small anisotropy c, the noise is expected to be important and induce tip splitting. Whcn the tip is destroyed, thc structure becomes N

25. Unidirectional Solidification from the Solution

6

8 1

:

FS,,'

2 2-:,'

2

101

,' FD

i coO(A)

9

CD

....."

Figure 24.16: Morphological phase diagram in the phasc space of the undercooling A and the anisotropy strength E . CS: compact seaweed, equivalent to DBM, CD: compact dendrite, FS: fractal seawced, and FD: fractal dendrite [34]. fractal for a short length scales, as is described in the subsection 23.2. These structures are called fractal seaweed (FS) and fractal dendrite (FD) in their morphology diagram, Fig.24.16.For thc details, refcr the original paper 1341.

25

Unidirectional Solidification from the Solution

An example in which the growth pattern is controlled by the material diffusion is the alloy crystal growth from solution. Solution of a binary alloy is cncapsulatc in a thin Hele-Shaw cell, a thin rectangular parallelepiped ccll (Fig.25.1). One end of the cell is kept hot and the other end is kept cold to impose a temperature gradient

Figure 25.1: Unidirectional crystal growth from solution in a Helc-Shaw cell. Solution is placed in a narrow gap bctween two glass plates, heated at one end and cooled at the other end. The cell is pulled steadily in thc cool region, facilitating thc steady growth of the crystal.

102

Part IV. Diffusion-Limited Growth: Pattern Formation

Figure 25.2: Equilibrium phase diagram of a binary alloy. The liquidus temperature TLdecreases from the melting temperature TMof the pure A material by increasing the concentration c of B atoms. over the sample. When the cell is pulled in the cold side, the solution is crystallized unidirectionally. Thc crystal-liquid interface lies normal to the pulling direction. Since the liquid is hot and the crystal is cold, thc latent hcat is rclcased in the cold crystal. In this case, the heat diffusion will not induce instability of the flat interface as explained in the section 22. But there is a slow material transport which controls the crystal growth. The solubilities in the liquid and crystal are different, as is shown in the phase diagrcm Fig.25.2. The component less soluble in thc crystal is expelled in the liquid and should be transported far away down by the diffusion. The other component, more soluble in the crystal should bc supplicd far away from the liquid by the diffusion. This material diffusion causes thc similar interface instability m the heat transport did in the melt growth, and leads @,tern formation with periodic structure. To summarizc, in the unidircctional growth heat conduction stabilizes thc interface and thc chemical diffusion causes the instability. Since the hcat conduction is faster than the material diffiision, and the heat conductivity in the crystal and liquid are almost the samc, onc often uscs the approximation that the thermal gradicnt GT is thc samc in both phases. When the samplc is sandwiched in a thin cell, the heat conduction is mediated by the cell wall, and the thermal conductivities in both crystal and liquid phases arc the same. Then the temperature distribution is simply describcd as T ( z )= To

25.1

+ GTZ.

(25.1)

Fundamental Equations

We consider the phase diagram of AB alloy as shown in Fig.25.2. By including B

25. Unidirectional Solidificationfrom the Solution

103

TO

Figure 25.3: Concentration distribution drawn in a heavy line at the vertical axis representing the temperature T and the height I simultaneously from the relation T = Ti GTZ.Here TO= TM- mc,/k.

+

atoms with a concentration c, the melting temperature decreases from that of the pure A atoms, TM.At a temperature T below T M the , crystal phase is stable when the concentration c is less than cs(T), and the liquid phase is stable when c is more than CL(T). Between these two concentration, cs(T) < c < C L ( T )the , crystal and liquid phases coexists. Phase separation between the crystal and the liquid phases takes place, and a crystal with a concentration cs(T) coexists with liquid with a concentration CL (T )The . liquidus line is well approximated by a line

TL= TM- mcL(T)

(25.2)

with a slope m. The partition ( or segregation) coefficient k is defined as the concentration ratio between the coexisting two phases as (25.3) and the miscibility gap is defined as

AC= C L ( T-) cs(T).

(25.4)

By pulling the two-dimensional cell with the solution of concentration em to the low temperature side with a constant velocity V, the steady crystallization takes place. From the global conservation of material, the concentration of the crystal alloy should also be coo, as denoted in the equilibrium phase diagram, Fig.25.3. First we consider the case when the surface is flat at I = 0. Since the material diffusion in the crystal is negligible, the concentration in the crystal should be em = cg. Assuming that the local equilibrium is satisfied at the interface, the liquid concentration at the interface is ct = c,/k and an interface temperature is mcm To = TM- k '

(25.5)

Part IV. Diffiision-LimitedGrowth: Pattern Formation

104

A conccntration difference is created in thc liquid at the interface cy, and far from there c., Thc conccntration variation follows thc diffusion equation (25.0) with DC.bcing thc chcxnical diffusion constant, and the first term on thc r.h.s. is due to the fixt that thc laboratory franir is comoving with the crystal with the velocity V. Whcn thc interface deforms as

z

(25.7)

= <(z,t ) ,

tlic teIripcratiirc at thc intcrfacc varies as

Ti = To

+ Gr(.

(25.8)

Since the Hclc-Shaw ccll is thin, wc can neglect variation of physical quantitics in tlic , ~liquid ~ dircctioiis of the thickness, ?/-direction. Tlic cquilibrium conccntration c ~ , of at thc interfxe tempcratmc is dctcrmiiicd from Eq.(25.2) with the Gibbs-Thomson ciirvaturc effect iiicluded as (25.9) with L bcing thc latent licat pcr volume. By assuming a local cquilibrium at tlic interfacc, cr,(x,<) = Cr,,c.r,l the liquid concentration at the interface is dctcrmined from Eq.( 25.8-25.9) ~LS cm WTM GT cr.(n:.() = - - -K - -<. (25.10) k 7n.L m From the local equilibrium assumption, thc concentration in thc: crystal c s ( 2 , <) is given by cs(.x,<) = kcI,(n:,<) with tlic cquilibriuin segregation cocfficicnt k. On crys() - cs(x,f,)) is cxpcllcd from the crystal per unit tallization, excess Inass xL(cI,(z, time, and this cxccss inass has to IIC transported by ciicmical tliffiision flux -D,&/an. The mass conscrvation reads as

(25.11)

Wc now introduce a dimensionless diffusion ficld in tlic liquid by %(Z,

c-cm z,t)=-

AC

C-Cc,

(k-' - l ) ~ ,'

(25.12)

which sntisfics thc far ficld condition u ( z ,z --t co)= 0. For a flat intcrface (C = ti = 0), ZL is 1 at tlic crystal-liquid boundary. Thc diffusion cquation rcduces to

(25.13)

25. Unidirectional Solidification from the Solution

105

with the diffusion length E D = 20,/V. When D, is very large, the diffusion field u relaxes to steady state &/at = 0 quickly during the slow growth of a crystal. The field u follows the steady state equation 2 au -+ v 2 u = 0.

(25.14)

lo dz

The local equilibrium condition (25.11) is written in this dimensionless form as

<

uq,= 1- dR - -.

(25.15)

IT

A chemical capillary length WTM d= -

(25.16)

mLAc characterizes the surface effect, and a thermal length 1T =

mAc GT

(25.17)

-

characterizes the scale of the temperature variation. Since the concentration of the solid at the interface is (25.18) ui,s = k(ui,L - I), the material conservation (25.11) is written as (25.19) Here V, is the normal growth velocity given by (25.20)

25.2

Stability of a Flat Interface

When the flat interface is growing steadily in z direction with a velocity V, the diffusion equation has a solution

uo(.z)= e-22/’n, which satisfies the boundary condition UO(Z = 03) = 0 and interface deforms to z = <(x,t ) = aqe+ cos qx,

(25.21) UO(Z = 0) = 1.

When the (25.22)

the diffusion field also deforms rn

u(e,z,t) = u&)

+ Alewqfcosqee-hqZ.

(25.23)

Part IV. Diffusion-Limited Growth: Pattern Formation

106

IW

Figure 25.4: Dispersion relation of the interface modulation during the unidirectional solidification. Here A1 is proportional to a,, and the damping rate in t direction, A,, should satisfy the diffusion equation in quasi-stationary approximation (25.14) as

or = 10'

+ Jm.

(25.24)

From the local equilibrium condition (25.15) A1 is determined as (25.25) By inserting these results in the material conservation (25.19), one obtains the dispersion relation W

DC

(I,'+Jm) (F-L-dq2)-(;) D

1T

2+ -2( l - l ~ ) ( 1 ~ + d q ~ ) , (25.26) 10

T

which is depicted in Fig.25.4. For slow pulling velocity V , W, is negative and the deformation decays in time. The flat interface is stablc. When the pulling rate V is larger than the critical value V,, the maximum value of wg becomes positive, and the flat interface becomes unstable against those deformations with wavenumbera with w, > 0. At the critical velocity the maximum of w, becomes zero: wq= dw,/dq = 0. (See Fig.25.4.) Since V, is small, the diffusion length E D = 2 D / V is large such that the relation ZD, IT >> d holds, and the wavcnumbcr for the most unstable mode is expected as large as q l D >> 1. Then the dispersion is approximated as

x:,

(25.27)

25. Unidirectional Solidification from the Solution

1

10

107

100

1000

10000

Figure 25.5: Neutral curve of the unidirectional solidification. The critical velocity V, is obtained as

-v,-

2

_ - N - N

1

+ -3-

Dc ~ D , c 1T 1D,c and the critical wavenumber is

4k2d

(

lD,c)

1

lI3 7Z

3 2k2d

lI3

1

1T -I-2 (T) G1

(25.28)

(25.29) For V > V,, wq is positive for some regions of wavenumbers, and the flat interface is unstable for these sinusoidal modes with wavevectors within this region. The locus of wq(q,V ) = 0 represents the neutral curve q = qN(V) as shown in Fig.25.5. The neutral curve is shown to be closed at the upper critical velocity Va.When the pulling velocity is too fast, the diffusion length becomes of the order of the capillary length, IT >> 1~ d , and the interface is stabilized by the capillarity. This is called the absolute stability. For q l D << 1, the dispersion relation is simplified as N

(25.30)

At the upper critical velocity (25.31) the maximum value of wq vanishes: wq = dw,/dq = 0. For the velocity higher than V,, wq always stays negative. The critical wavenumber is given by qa O:

[k(l

+ 2k)d31T]-”4.

(25.32)

Part IV. Diffusion-Limited Growth: Pattern Formation

108

25.3

Eckhaus Instability

+

When the interface deforms sinusoidally as z = [(x,t ) ['(x, t ) = J Re(a,e*g")dq, thc Fourier transform a,(t) devclops as

(25.33) in the linear approximation. Here w, is dctcrmined from thc dispersion relation (25.26). When the pulling velocity V is a little larger than the critical value V,., the interface is unstablc for the sinusoidal modcs with w,, > 0. Around the critical wavclength qc, w,, is cxpandcd as wq

= wc - C(q - %-)2

(25.34)

with w, = w,,, and C = -;(a2w/aq2),, is positive. wc is proportional to V - V,. From Eq.(25.33) and (25.34), thc timc evolution of thc intcrfacc ((x,t ) can be approximated

as (25.35) Near the critical point V,, thc unstable modc has tlic wavc numbers around qc, and the deformation is cxprcssed as

< = A(x,t)e"lcl'

(25.36)

with thc slowly varying complcx amplitudc A ( s ,t ) . It satisfics tlic lincar equation

(25.37)

As w, is positive, the deformation A incrcascs, but thcn the nonlinearity should come into play. Since the system is invariant by the transvcrsal translation x + 2 + 4 and the space inversion 5 -+ -5, one gets thc nonlincar amplitudc cquation of tlic Landau-Ginzburg type, (25.38) There is a systematic derivation of the Landau coefficicnt CYI by thc reductive pcrturbation mcthod [199]. To spccify thc mcaning of thc Landau cocfficient a l , wc consider the amplitude of the critical modc: A ( x , t ) = A(t). If a I> 0 the third order term acts as to limit tlic amplitude A. Thc amplitudc increases gradually as A wA/2 (V - Vc)'/' from xcro near thc critical point, as shown in Fig.25.6. The transition is similar to the second order phase transition in cquilibriiim case, and is called supcrcritical. If ct1 < 0, om nccds still highcr order t e r m to obtain a finite amplitude of deformation. For cxamplc, with thc fifth-ordcr term A5 with negative N

N

25. Unidirectional Solidification from the Soh tion

109

Figure 25.6: Schematics of the modulation amplitude of the interface near the instability. For a positive Landau Coefficient 0 1 , the bifurcation from the planar to the modulated interface is supcrcritical, whereas for a negative cyl it is subcritical. coefficient, the amplitude jumps to a finite value at V , with thc hysteresis. The transition is similar to the first-order phase transition in equilibrium, and the bifurcation is called subcritical. We consider hereafter only the supcrcritical case with al > 0. One can easily find that there is a stationary solution to Eq.(25.38) in the form of a single sinusoidal mode as

&(.)

= A,c'(*--"c)".

(25.39)

Its amplitude A, is determined as

(25.40) In the domain of w, > 0 where the flat interface is unstable, a stationary profile of the interface is possible which is deformed sinusoidally as t = Rc(A,e*'") with p dependent amplitude A,. But what periodicity is selected? Are all these nonlinear solutions stable? We consider now thc linear stability of the stationary state against the modification: A ( s , t )= Aqe*(Q-qC)" (1 [(s,t)] (25.41)

+

with a complex modification [. For small (, the modification can be written as 1 6 ec eRc%*Imc.This means that the real part Re[ represents an amplitude modification to A,, and the imaginary part Im< represents a phase modulation. By inserting Eq.(25.41) to Eq.(25.38), and taking the first ordcr of [, one gets

+

N

N

(25.42) This linear equation can be analyzed by using the Fourier transformation as

Part IV. Difftision-Limited Growth: Pattern Formation

110

with real B1 and B2. By comparing coefficients for e'Qz and c-'@, we get the linear equation for B1 and B2 as

RBI = C [-2Q(q - qc) - Q'] Bi RB2 = C [2Q(q - q c ) - Q'] B2

[W - C( q - qC)']

+ B2) + Bz).

(BI

- [u.- C(q - qc)2](&

(25.44)

Here is replaced by the corresponding term by Eq.(25.40). On requiring that B1 and Bz has nontrivial solution, the eigenvalue equation

+ 2CQ(q - + CQ' +

W, - C(q - qc)' wc - C(q - qr)2 R - 2CQ(q - q , ) + CQ' +w, - C(q - qc)2 1 = a' + 2C [Q' + Q$ - ( q - qr)'] R + 2C2Q2[Qk - 39' + 5(q - qC)']

qc)

W, - C(q - qc)'

= o

(25.45)

is derived. Here Q:, = w,./C. Eigenvalucs arc obtained as

fh= -C [QZ - ( 4 - qc)'

+ Q'] f cJ[Q$- ( 4 - qr)']]"+ 4Q2(q - qc)2.

(25.46)

For a long wavelength modulation with a small wavenumber Q, one gets

R-

x -2C [QL - ( q - qc)2]+ O(Q2)x -2w, (25.47)

with corresponding cigenmodes; B2,- = B1,- = B- and Bz,+ = -BI,+ = B+, or [- = B - 6 - l cos Qx and [+ = iB+e"+'sin Qx. Therefore, - mode corresponds to the amplitude modulation and mode corresponds to the phase modulation. Since R- < 0 for w, > 0, amplitude modulation always damps out. On the contrary, R+ can be positive for those modes with wavenumbers Q in the region

+

(25.48) The phase modulation destabilizes the stationary solution. Within the region (25.48) near the neutral line q = q N ( V ) , the phase diffusion coefficient is negative and thc phase diffusion mode increases. This is called thc Eckhaus instability [55]. Neutral curve q = Q N ( Vis) determined from the condition w, = 0. With Eq.(25.34), then, qN,f = qc f @. From Eq.(25.48) the Eckhaus boundary is obtained as qE,-+ = qc f &(qN,+ - qr), as depicted in Fig.25.7. In the analysis the dispersion rclation, Eq.(25.34), is assumed to havc a maximum at the critical wavenumber qc even in the supercritical regime V > V,. In reality, the most unstable wavcnumber qm shifts rapidly depending on V. Therefore, the approximate Eckhaus boundary can be written around the most unstable mode as qE,+ = qm f $(qN,+ - q,,,). Brattkus and Misbah calculated the Eckhaus boundary more precisely, which is quite different from parabola [as].

25. Unidirectional Solidification from the Solution

111

Figure 25.7 The sinusoidal modulation with wavenumbers q inside the region E is unstable against the phase diffusion, and this is called Eckhaus instability.

25.4

Fully Nonlinear Behavior

Near the critical point where the surface deformation is small, the amplitude equation (25.38) with a1 > 0 describes the crystal shape correctly. When the Landau coefficient 01 is negative, or the system is far from the critical point, the deformation is no more small and the full nonlinearity has to be considered. In such a case, numerical simulation is appropriate to study the large interface deformation “3, 106, 1671. We extend the simulation algorithm explained in Appendix A24.3 to the periodically modulated surface structures under a constant temperature gradient. For a small velocity near the critical velocity V,, the cellular structure appears as shown in Fig.25.8a. On increasing the pulling velocity, the cell groove deepens and the surface takes the form of cusp axrays, as in Fig.25.8b. Further increase of the pulling velocity sharpens the tip and the dendrite array is formed as in Fig.25.8~.Here the dendrite structure is controlled by the anisotropy of the surface tension and the tip r a d i u s p d the velocity satisfies the solvability condition (24.5). The simulation results qualitatively agrees with the experimental sequences of morphology changes as shown in Fig.25.9.

Figure 25.8: Interface structures in unidirectional solidification obtained by simulation: (a) arrays of cells (V/V,= l.Ol), (b) of cusps (V/Vc= 1.15), and (c) of dendrites ( V / K = 17.6)

Part IV. Diffusion-Limited Growth: Pattern Formation

112

Figure 25.9: Interfacc structures in unidircctional solidification of succinonitrile in acetone. (a) Arrays of cells, (b) of cusps and (c) of dendrites [180].

t

Figure 25.10: Tilted iinidircctional solidification in thc pivalic acid-ethanol system. Crystal axis is .ct, = 40.5" . Thc velocity in pm/s are (a) 0.5, (b) 1.0, ( c ) 2.75 and (d) 10.0

[m].

25. Unidirectional Solidificationfrom the Solution

113

(c)

I

I

V/V,=6.12

I

Figure 25.11: Tilted unidirectional solidification at various pulling velocities obtained by simulation. The crystal axis is $ = 17" off from the direction of the temperature gradient. The tilt angle 4 of the tip profile is (a) 4.3", (b) 8.7", (c) 12.5", (d) 14.2" and (e) 16.7" [148].

In directional solidification there are two tip stabilizing effects, the thermal gradient and the surface stiffness. What will happen when two anisotropies prefer different orientations: the temperature gradient forces the crystal to grow in the z direction, but the capillarity prefers mother direction with a tilt angle $. It means that the capillary length depends on the orientation II, of crystal axis, and is written as d = d(l [l - €cos4(8- $)]

(25.49)

at the point with the angle 8 of the interface normal to z axis. Tilted unidirectional solidification is observed experimentally [73, 182, 27, 2, 1441. Figure 25.10 shows interface structures at various pulling rates for the pivalic acidethanol system 11821. On increasing the pulling rate, the interface tilts from the direction of the temperature gradient, and approaches to that of the crystalline axis. Since the tilting increases by increasing the velocity, the surface kinetics is supposed to be relevant. Also from the linear stability analysis, the anisotropy in the surface stiffness is found to be irrelevant for the tilting [51, 2021. But the recent numerical simulations of the unidirectional solidification with local equilibrium assumption [2, 1481 show that the tilting is possible when the crystalline axis is off-angled from the pulling direction. The simulation in Fig.25.11 shows that for small pulling velocity V the cell tip is oriented close to z axis, the orientation of temperature gradient, but for large V the dendritic crystal is oriented close to the crystalline axis. Thus

Part IV.Diffusion-Limited Growth: Pattern Formation

114

the experimental tendency of large tilt angle at fast pulling can be reproduced by the surface stiffness, too. More studies are necessary to identify the real mechanism of crystal tilting, whether it is due to the anisotropy in capillarity or to that in the kinct ics.

26

Eutectic Growth

For some alloy systcm such as Pb-Sn, the phase diagram looks as shown in Fig.26.1. Due to the mixing entropy the melting temperature of an alloy often decreases from the pure material. When two liquidus lines of AB alloy decrease down from the pure A and from the pure B melting temperatures by mixing the other component, they meet at some concentration of B species CE and the temperature TE. At TE, the liquid solution with the concentration CE coexists with two crystals, phase cx with c ~ ) This . triple point is concentration cg(< CF,)and phase p with concentration t$(> called the eutectic point. At this eutectic point, the melting temperature is minimum.

If one grows this eutectic alloy unidirectionally in a Hele-Shaw cell, what kind of structure appears'? Starting from the liquid with the eutectic concentration CE, crystallization of a or p phase alone cannot satisfy the material conservation. a and phases should appear simultaneously. It may intuitively expected that a and p phases appear alternatively as lamellae parallel to the growth direction: When the crystal is growing in positive z directions, the symmetry in 5 and -5 directions yields that thc a@phase boundary is expected to align in z direction and perpendicular to the liquid-crystal interface in I direction. This is the structure found in experiment (911 at low pulling velocity, as shown in Fig.26.2 (1081. The problem to be posed is the selection of the periodicity A. There is a detailed calculation by Jackson and

p

T

Figure 26.1: Equilibrium phase diagram of aa eutcctic AB alloy.

26. Eutectic Growth

115

Figure 26.2: Lamellar growth of eutectic crystal, CBr4- CZC16 [lo81 Hunt, which is reproduced in Appendix A26 [91].Here a phenomenological discussion is given on the periodicity selection.

26.1

Lamellar Structure

Since the thermal diffusivity is larger than the chemical diffusivity, the temperature gradient GT = dT1d.z can be assumed constant. By choosing the origin of z axis at the point where the temperature is at the eutectic value TE,the temperature distribution is described as T ( z )= TE GTZ. (26.1)

+

The liquid ( z > 0) is hot and the crystal ( z < 0) is cold. We consider the simplest case that a crystal is growing from a solution with eutectic concentration CE. When the crystal takes a lamellar structure with a periodicity A, the ratio of a! phase q and that of p phase 1 - 77 should satisfy the conservation relation CE

+ (1 - q)4

(26.2)

4-

(26.3)

= qcg

or

v=with the miscibility gap

Ac =

CE

Ac

4-

C;

(26.4)

Since @ is smaller than CE, B atoms are expelled out from the a crystal into the liquid by crystallization, and the liquid concentration c; in front of a! phase increases above CE. On the contrary, in front of the p phase, B atoms are sucked up by p crystal and the liquid concentration cf decreases below CE during crystallization. As is apparent from the magnified phase diagram Fig.26.3 with metastable liquid branches, deviation of the liquid concentration from the eutectic value means that the supercooling at the interface under the local equilibrium assumption. The crystal concentration cg and

Part IV. Diffusion-Limited Growth: Pattern Formation

116 7

TE T E - A7

Figure 26.3: Phase diagram near the eutectic temperature TF,with metastable branches of coexistence curve. correspondingly differ from those at the eutectic temperature, but their differences are small and do not modify the conclusion essentially. 00 Since the concentration in front of each crystal is different from that at J or CE, the concentration diffusion takes place. The diffusion flow compensates the material deficit or surplus produced by the crystallization. In front of the 01 crystal, v,(cE - cg) M V,,(CE - cz) of B atoms are expelled from the unit surface per unit time. This excess is transported by the diffusion flow -D,ac/dn. However, the material does not need to be transported up to z + 00: It is sufficient to be transported only to the neighboring lamellac of 0 phase, where thc material is deficient. Since , phases is cE - cf for a the concentration difference in liquid in front of the a and 9 separation of X/2, the growth rate is determined as --f

(26.5) If the interface is flat, the direction of material diffusion x is orthogonal to the growth direction .z and the diffusion docs not contribute to the growth. Actually, the interface is curved, and the contribution to growth appears. This is taken into consideration in Eq.(26.5) by the factor Qn. Its detailed form is explained in Appendix A26. In consideration of the material conservation in front of the 0 phase, we get the similar formula with coefficient Q p . In the stcady state where thc growth velocity of a and p phases are the same, the coefficients should satisfy the relation (1 - q)/Q, = q/Qp due to the conservation (26.2). The liquid-crystal interface is curved in order to satisfy the mechanical force balance at the triple point, where three phases, liquid, 01 crystal, and /3 crystal phases, meet (Fig.26.4). The force balance in x and z directions is described as: TI,,

sin 8,

+ YLO sin 6'0 = 7-0

cos e, = Yr,O cos ea.

(26.6a) (26.6b)

Here YL, is the surface tension between the liquid and the a phases which is assumed isotropic for simplicity, and so on. In equilibrium the liquid-a ( p ) crystal interface is

26. Eutectic Growth

117

Figure 26.4: Interface profile near the triple point for the cutectic growth. part of a circle with a radius R, (Rp),and the angle 8, ( 8, ) is related to R, (Rp ) and the periodicity X as 2Rasin8, = vh,

2RpsinBp = (1 - q)X,

(26.7)

as is evident from Fig.26.4. The interface temperature is given by thc concentration deviation from the cutectic values and also by the Gibbs-Thomson curvature effect as Yl.aTF, 1 T o = TE- m,(cE - cE) - -L, R,

(26.8a) (26.8b)

Here ma = IdT,/dcTl and ma = IdTp/dgl are the absolute slopes of thc liquidus lincs for a and O, phases, Lo, La are latent heats of a and p phascs, respectively. By adding Eq.(26.8a) and (26.8b), liquid concentration diffcrcncc is written as

Here the interface supercoolings are defined as AT" = TE- T" and AT* = TE- TP. We consider the simplest casc where crystal phases 01 and p arc symmetric: cF, = 112, ma = mp = m, 8, = Oa = 8, q = 1/2, Qn = Qp = Q , yLn = Y I , ~= y: AT" = ATp = AT, and c: - cg = Ac/2. By using the capillary length d = yr.,TE/LmAc, the interface supercooling is written from Eqs.(26.5) and (26.9) as (26.10) which consists of two contributions; Diffusional one ATD= TEanX/ln and the kinetic one AT, = T E a K d / X . Here ln = 2D,/v is the diffusion length, and material paramcters are represented as U D = m n c / 4 Q T ~aK , = 4sin BmAc/TF,.The relation bctween A T and A is shown in Fig.26.5.

Part IV. Diffusion-LimitedGrowth: Pattern Formation

118

h Figure 26.5: Minimum undercooling hypothesis for the selection of the eutectic lamcllar periodicity. In order to determine the lamellar periodicity A, thc minimum supercooling hypothesis is used by Jackson and Hunt: aAT/aA = 0. Then the periodicity is determined as (26.11)

-

This yields the relation between the pulling velocity v and the periodicity A,

as (26.12)

Also the interface undercooling AT,,, is determined as

Figure 26.6: Variation of the average interlamellar spacing with the growth velocity w for the carbon tctrabromide (CBr4)- hexachlorocthenc (CzC16) cutectic system [174].

26. Eutectic Growth

119

(26.13) and it satisfies the relation with the pulling velocity 2) as

AT: V

2sind

- Q

d ( r n A ~ ) ~ -= const.

(26.14)

D C

The full nonlinear analysis by computer simulation [107, 95, 98, 961 also show that the interface undercooling has the minimum as a function of the periodicity, and that the Jackson and Hunt theory provides the accurate position of the minimum undercooling [96]. The scaling relations, (26.12) and (26.14), are also shown there [98, 961. These relations are also confirmed by experiments, as shown in Fig.26.6 [174, 2031.

26.2

Parity Breaking and Oscillation

On increasing the pulling velocity V or on varying the liquid concentration from the eutectic value, the lamellar structure is found to be modified. For slow pulling rate, the phase boundary between o and @ phases align parallel to the temperature gradient. On suddenly increasing the pulling rate by a factor 4, the periodicity selected by the minimum undercooling condition should be half of the initial one, as Eq(26.12) tells. But such a large structural variation is not possible topologically. The numerical simulation using the boundary element method in the stationary code found that the parity (left-right symmetry) will break for large pulling rate [52,95,97]: The LYPor @a! triple point shifts transversally along the liquid-crystal interface, and the a@phase boundary tilt from the temperature gradient, as shown in Fig.26.7a. The tilting is found later in many experiments (Fig.26.7b) [57, 581.

fa)

( b)

Figure 26.7: Parity broken lamellae in eutectics obtained (a) by simulation and [97] (b) by experiment of CBrd-C2C16 [57].

Part IV. Diffusion-Limited Growth: Pattern Formation

120

Figure 26.8: Periodic oscillation of lamcllae in off-eutcctic alloy (a) by the experiment of Al-CuA12, and (b) by simulation [204]. This tilting is different from that discussed in the previous section 25.3. There the tilting is enforccd by the tilting of crystalline axis from the growth direction and the parity is externally broken. Herc the system has the left-right parity symmetry, but the realized state breaks the parity spontaneously. , shows By shifting the far ficld concentration cm from the eutectic value c ~system the nonstationary structurc, as pcriodic oscillation shown in Fig.26.8a [204]. The linear stability analysis showed the instability of the oscillatory motion of the triple point (53, 941. There is a model simulation which shows the oscillatory bchavior, and the result is in good agrccmcnt to thc experiment as shown in Fig.26.8b [204].

27 Diffusion Effect on Polyhedral Crystal: Berg Effect So far it is assumed that thc surface kinctics is infinitely fast and the local equilibrium is realized at the interface in order to strcss tlic diffusion effect on thc front instability. This approximation is not far from reality for an ice crystal growing in water [65, 661. The prism facc of ice in water is rough, and the heat transport controls the growth. On the other hand, for snow growing from thc water vapor, the approximation does not hold since thc surface of the snow is sharp and faceted. The kinctics is slow and has to be properly taken into account. If the interface is atomically rough, the kinetic coefficient K in Eq.(18.8) is finite but not singular. The growth shape can be anisotropic but is nonsingular and smooth. Thc orientation dcpendcnce of K leads to the ~ i c wscaling relation of the dendritic growth rate and the tip radius [32, 33, 50, 169).When thc prcferrcd oricntations of the surfacc tension and of the kinetic coefficient are diffrrent, rich variety of growth shapcs with dynamical shapc transitions is expected [45]

27. Diffusion Effect on Polyhedral Crystal: Berg Effect

121

Figure 27.1: (a)Electrostatic potential around metallic polygon. The metal surface is equipotential. (b) Concentration field distribution around the square crystal growing steadily with a constant normal gradient q, = &/an along the surface.

If the interface is atomically flat, the kinetic coefficient K is singular, and the crystal is expected to take polyhedral morphology covered with flat facets of singular surfaces. The growth of these singular surfaces is governed by the surface kinetics discussed in Part 111. However, material has to be transported to the growing interface by the diffusion, and the diffusion also influences the growth form. One has to consider both the kinetic and diffusion processes in this situation. First we consider the concentration distribution around the polyhedral crystal growing from the solution. It is determined by Eq.(18.6-18.8). Usually, crystal growth rate is small for kinetic-controlled growth, and one can assume the stationary condition that the diffusion field quickly adjust its distribution around the given crystal morphology: au/& = 0. Thus the diffusion field u satisfies the Laplace equation

v2u= 0

(27.1)

instead of the diffusion equation (18.6). An analogy holds between the concentration distribution u(r) and the electrostatic potential q5(r). Far from the crystal, the concentration distribution reflects the isotropic nature of the system and equiconcentrations are asymptotically spherical. If the concentration is assumed constant on the polyhedral face, u(r) corresponds to electrostatic potential q5(r) around the metallic polyhedron, and looks as shown in Fig.27.la. Near the corner of the polyhedron, the spacing between consecutive equi-concentrations are narrow, and the normal growth rate determined by Eq.(18.7) or V, = -D,dc/dn is larger at the corner than at the center. Then the polyhedral face cannot remain flat. To keep the polyhedral

122

Part IV. Diffusion-Limited Growth: Pattern Formation

Figure 27.2: Concentration distribution around the growing NaCl crystal from water solution [24]. face flat, there should be concentration difference on the surface, especially between those at the corner and at the center. If we assume the constant normal gradient on a flat but bounded surface, the concentration at the corner should be higher than that at the center, as shown in Fig.27.lb. This was shown analytically in two dimensions [173]. Experimentally, the concentration profile around the polyhedra3 crystal is observed by Berg as shown in Fig.27.2, and the concentration difference along the surface is observed. This effect is called the Berg effect 1241. The distribution of the supersaturation along the flat surface is also measured (421. If the face is redly flat and the concentration varies on it, the kinetically controlled growth rate (18.8) varies from position to position, and it is impossible to keep the face flat. To realize a steady growth, the kinetic coefficient should vary along the surface. This is possible when a macroscopically flat surface contains many microscopic steps and is in fact a vicinal facc. Then the kinetic coefficient depends on the concentration. For example, at the corner of the crystal where the supersaturation is high, the two-

Figure 27.3: Formation of snow dendrite by simulation [201].

27. Difision Effect on Polyhedral Crystal: Berg Effect

123

dimensional nucleation supplies steps toward the center of the face [119]. The step advancement velocity v decreases new the center where the supersaturation is low, but if the step density is high and the step separation 1 is small, the normal growth rate R = avll can be kept constant over the surface. If the supersaturation at the corner is too high or the crystal has grown too large, the difference in supersaturations at the corner and at the center becomes too large to keep the constant normal growth rate over the flat face. The corner starts to grow faster than the center, and the instability sets in [47,1191. This initiation of the corner instability is analyzed numerically, and the corner instability and its size dependence is found, if the nucleation controls the kinetics 11191. Also by using the boundary element method the growth of a kinetically-controlled two-dimensional crystal is simulated. The circular crystal becomes polygonal, and then produces the faceted arm from the corner of polygon, similar to the snow growth, as shown in Fig.27.3 [201]. For the sidebranch formation of a faceted dendrite, however, more studies seem necessary.

Part V

Appendices A9 More on Surface Roughening A9.1 Solid-on-solid (SOS) model In order to focus on the temperature dependence of the surface structure, the solid-onsolid (SOS) model is often used [112,43]. The model picks up the freedom of surface height z ( i ) such that no vacancy in the crystal, no overhangs at the surface and no crystal clusters in the vapor arc allowed. Here i is the site in a d-dimensional hypercubic lattice with 2d nearest neighbors, and the crystal occupies d 1 dimensional space. In a unit of an atomic size a, the height h(z) = z ( i ) / a takes integer values, h(i) = -00,. . . ,- l , O , 1 . . . , 00. Since the height difference between neighboring sites costs energy, the Hamiltonian 8 is written as

+

(A9.1) Here the model with p = 1 is cdled the (absolute) SOS model, the model with p = 2 the discrete Gaussian (DG) model. The latter is convenient to treat analytically. If the height takes only two values, h ( i ) = 0,+1, the model corresponds to the Ising model. As to measure the surface roughness, the height difference correlation function

PI

G(rij) = ( ( h ( i )- h(j))2)

(A9.2)

is used. When the surface is flat, heights arc strongly correlated and G remains finite even for r+ --t 00. When the surface is rough, heights at different positions fluctuate independently, and G(r + 00) diverges. In an Ising model, G can at most be 1, and the surface can never be rough in this sense.

A9.2 Monte Carlo simulation of the SOS model This appendix contains a source list of an example program of Monte Carlo simulation explained in Section 9.1. It is hoped that it provides a first step for the reader to get familiar with the simulation and to develop one’s own program. Random number generator is copied from the reference [67]. program so8 c============================================

c

Monte Carlo simulation of 2D-80s model for roughening

=----------------------------------------------_-_________--__-__------------------------parameter (lxmx=128,lymx=128)

A9 More on Surfse Roughening

dimension ih(lxmx,lymx) ,prob(6) read(*, *) lx,ly,loop,temp,field,iseed c----"---------'------------------------------------------------

c ih(ix,iy); solid height c lxmx,lymx; maximum linear size in x- and y-direction c lx,ly ; actual linear size of the system c prob(k) ; transition probability C k=l f o r crystallization C k=2-6 for desorption uith k-2 neighbours c temp; temperature, (coupling J=I) c field; chemical potential difference c loop; loop number f o r small accumulation and averaging c iseed; seed of the random number c mag; magnitude of height sum c nnex; nearest neighbor energy c----""----------"----------------'--

call ranint(iseed1 call init(lxmx ,lymx, Ix,ly ,ih) write(*, *> 'lx,ly ,loop,temp,f ield ; ' write(*, *) lx,ly,loop,temp,field write(*,*) 'ilp,mag.nnex' call engy (lxmx,lymx, lx,ly ,ih .mag,nnex) call prbini(prob,temp,f ield) c-------------- monte carlo loop do 10 ilp=l,loop do 11 inlp=l,lx*ly call mc(lxmx,lymx,lx,ly, ih.prob,mag,nnex) 11 continue write(*, *)xilp,mag,nnex 10 continue c----------- check the final height and exchange energy call engy(lxmx,lymx,lx,ly,ih,magl,nnexl) write (*, *) mag,magl ,nnex,nnexl stop end C""=e=============rra==============t=============

subroutine init(Ixmx,lymx,lx,ly, ih) c---"'-----------""-----------------------------------------C

initiallization

c------------------'-"------------"-----------------

dimension ih(lxmx,lymx) do 200 iy=l,ly do 200 ix=l,lx

125

Part V. Appendices

126

ih(ix,iy)=O 200 continue return end ......................................................

subroutine mc(lxmx, lymx,lx,ly ,ih,prob,mag,nnex) t-----------------------------------------------------C Monte Carlo step t--------------------------------------------------------

dimension ih(lxmx,lymx) ,prob(6) isum=O ix=randmO*lx+l iy=randm() *1y+l ixp=ix+l ixm=ix-1 if(ixp.gt.lx) ixp=ixp-lx if(ixm.lt.1) ixm=ixm+lx iyp=iy+l iym=iy- I if(iyp.gt.ly) iyp=iyp-ly if (iym.lt . I) iym=iym+ly if(ih(ixp,iy>.lt.ih(ix,iy)) isum=isum+l if (ih(ix,iyp) .It.ih(ix,iy)) isum=isum+l if(ih(ixm,iy).lt.ih(ix,iy)) isum=isum+l if(ih(ix,iym).lt.ih(ix,iy)) isum=isum+l trpr=prob( 1)+prob(isum+2) trtr=randm() c----------------- adsorption if(trtr.lt.prob(1)) then ih(ix,iy)=ih(ix,iy)+l j bup=O if(ih(ixp,iy).lt.ih(ix,iy)) jbup=jbup+l if(ih(ix,iyp) .lt.ih(ix,iy)) jbup=jbup+l if(ih(ixm,iy).lt.ih(ix,iy)) jbup=jbup+l if (ih( ix,iym) .It.ih(ix, iy)) jbup=jbup+l mag=mag+l nnex=nnex+2*jbup-4 return endif c----------------desorption if(trtr.lt.trpx-) then ih(ix,iy>=ih(ix,iy>-i mag=mag- 1

0

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I 0 1 1 I G I l 1 Y l * l l I 09 I tJ- tl 1 % 1 s II %

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Part

128

V.Appendices

ip=mod( ( j-I)*32,521)+1 m(j)=O do 30 i=1,31 ii=mod(ip+i-2,521)+1 m(j)=2*m(j)+(ia(ii)-1)/(-2) ij=mod(ii+488,521)+1 30 ia(ii)=ia(ii)*ia(ij) ii=mod(ip+30,521)+1 ij=mod(ii+488,521)+1 20 ia( ii)=ia(ii)*ia(ij) j=O return end

-__-_______----------------------

c---------------------------------

function randm0 common /rand/ m,j dimension m(521) ip=521 iq=489 j=j+l if(j.gt.ip) j=l k=j+iq if(k.gt.ip) k=k-ip m( j )=ieor(m(k) ,rn( j)) randrn=m(j)*0.4656612e-9 return end

A9.3 Continuous Gaussian model If the height variable h ( i )does not take thc discrcte integer values but takes continuous values, and the energy is givcn by (A9.1) with p = 2, the model is called the continuous Gaussian model. The partition function and correlation function of thc model can be calculated straightforwardly. The partition function is defined as (A9.3)

One can introduce the Fourier transformation as (A9.4)

A9 More on Surface Roughening

129

with wavenumbers q = 2 n n / L and n = - ( L - 1 ) / 2 , . . ,0,1,. . . ,L/2. Here L is the linear dimension of the system such that N = Ld. The coefficient h(q) is given by +

1

C eiqrih(i).

h(q) = f

i

(A9.5)

i

In terms of h(q), the partition function is written as

Here Go(q)is the lattice Green's function defined by (A9.7)

where 6 is a vector connecting nearest neighbor sites. Surface free energy is obtained (A9.8)

In the last equality, the summation is replaced by the integration for a large system size N as C, = N ,J: ddq/(27r)d, The height difference correlation function is calculated for large r as [115]

[?

r2-d - 1

1

2d-1nd/21'(d/2)2-d ln r const

+ const + O(r'-d)

+

for d # 2 (A9.9)

for d = 2

,

where I'(d/2) is the Gamma function [l]. The d-dimensional surface with d 5 2 is always rough because the height difference correlation function diverges for two separate points, r -+ 00. On the contrary, for d > 2 the surface is always smooth.

A9.4 Variation approximation Continuous Gauss model shows that the surface is rough in two dimensions. In the real crystal with d = 2 , the surface can be both smooth and rough. The discrepancy is due to the neglect of the discreteness in the surface position in the continuous Gaussian model. The height should be quantized in unit of atomic size a. For analytical treatment on the phase transition and critical phenomena, a modification is introduced in the model Hamiltonian as 3.1 = J

x



[h(i)- h(j)I2- r J ~ c o s ( 2 7 r h ( i ), ) i

(A9.10)

Part V. Appendices

130

c

O

a

-

X

2

Figure A9.1: Convexity of exponential function. where the height variables h(i) are continuous, but the additional one-body potential with a small positive coefficient U prefers the height to take integer values. We here study the phase transition of the model (A9.10) by means of the variation method 159). The theory depends on the convexity of the exponential function; e-(a+b)/2< ( e ~ " + e - ~ ) / 2This . inequality is obvious from Fig.AS.1. When the stochastic variable z is distributed according to the probability Pr(z), the average (5) is given by (z) = J Pr(z)zcdz,and the inequality is generalized to e-(z) 5 (e-.). For a thermodynamic system with a Hamiltonian Bo, the canonical probability is defined bv (A9.11) with (A9.12) When the true system is described by the Hamiltonian

X=Xo+V,

(A9.13)

one can show that

Therefore, the true free energy F satisfies the following inequality F 5 Fo

+ (V)o

3''.

(A9.15)

If we make an appropriate choice of 7& which minimizes F * , then it should be a good approximation to the true free energy F . In order to make the configuration

A9 More on Surface Roughening

131

summation tractable, we choose the effective Hamiltonian ‘Flo as (A9.16)

where q-dependent parameters G(q) are to be determined to find a minimum F*. With this choice of ?lo, the probability Pr given by EQ.(A9.11) is Gaussian with (h(q))= 0 and (lh(q)l2>= G(q)* FO is calculated ~ t r ~ ~ ~ t f o r and w ~isd equal l y to fA9.8) with the replacement of Go(q) by G(q). ( ‘ F t o ) ~is simply N ~ B T /In ~ .(H)o, the average of the first term is easily calculated as

Since there axe only first and second order cumulants for a Gaussian distribution, the term containing the one-body potential can be calculated by using the relation

[

(coa27rh)o = Re(eZTih)o = exp 22n(h)o- -(h2)0] (2nY 2 with W2)O

=

1

1 ~(lhql”0 = iJ 4

= exp [-27r2(h2)o] (A9.18)

c

G(q).

(A9.19)

P

The approximate free energy F’ is obtained as

(A9.20)

If one calculates the higher order term (V”)o, it is proportional to U ” , and the present approximation is good for a small perturbation U . To minimize, F’ is differentiated by G ( d 88

Since the last term is independent of the wave number q, the q-dependent coupling G(q)has the form 1 K (A9.22) G(q) = Go(q)-l + K-11-2 ___ 42 t - 2

+

132

Part V. Appendices

0.4

0.3 y 0.2 0.I

0 0

0.1

0.2

0.3

0.4

X

Figure A9.2: Determination of the correlation length at various temperatures t = 7 r k ~ T / 4 J . where K-' = 2J/ksT, and the correlation length

< or the parameter z =

(T/<)~

< is determined as (A9.23)

By denoting x = ( T < ) - ~ ,t = 7rhF/2 = 7 r k ~ T / 4 J ,Eq.(A9.23) is transformed as

(")'

= f(x). (A9.24) J l+x Two curves y = z and 9 = f(z) are depicted in Fig.Ag.2 at various temperatures t . The intersection of these two curves gives the solution of Eq.(A9.24). The slope of f(s)at x = 0 changes drastically at t = 1, since z=

(A9.25) For small U and t > 1 there is only a trivial solution x = 0, but for t < 1 there is another nontrivial solution near x 0. For small x, the solution is approximated as N

For t < 1 the free energy F* for z # 0 is smaller than that for x = 0, and the state with a finite ( is stabler than the state with = 00. Therefore, the phase transition takes place a t t = 1 or 4 k ~ T k= -J. (A9.27)

<

7r

A9 More on Surface Roughening

133

The height difference correlation function is calculated by

ksT In [min(T , E)] . (A9.28) 2aJ Above the roughening temperature TR = 4J/nks or t > 1, there is only a solution fi:

x = 0 ( or ( = oo), and the height fluctuation diverges at T -+ 00: The surface is rough. On the other hand, for T c: TRor t < 1, E is finite and the height correlation function saturates as (A9.29) for r --+ 00. Below TRthe surface height fluctuates within a range of 6. This means that the steps of a length of order 6 can be thermally generated. The step energy 0 is then of the order kBT/( as kflT C -@-----exp (A9.30) t TR-T]. It approaches zero as well as its all derivatives by T on approaching the roughening point T --+ TR: It has an essential singularity at TR. For T > TR the correlation length 6 = 00 and 0 = 0.

I

A9.5 Renormalization group theory on roughening In the variation method (Appendix A9.4), only the lowest order of perturbation U is considered. The renormalization group method takes the cffect of higher order perturbation systematically [113, 114, 145, 1461. By coarse graining the two-dimensional space, onc gets the Hamiltonian in a continuum space limit as

- Vcos(2ah)

8 = J ' d x d y [ ~ ( V Z-) ~

(A9.31) Here the lattice parameter a is notified explicitly, and the height Z ( Z , y) is normalized as h(z,9 ) = ~ ( zy)/a. , L. is the linear dimension of the system. Coupling constants in (A9.31) are related to the microscopic parameters in (A9.10) as follows: the surface stiffness is here denoted by y = 2 J / a 2 and the strength of the one-body potential by V = U/a2. The height h(r) is expanded in Fourier series with wavevectors q between the lower, qc = n / L , and the upper, A = n/a, cutoffs.

h(r) =

=

C q,
h(q)e'q'=

h<(r) + h'(r).

C q,
eiq4'h(q)+-

C

e'P'h(p)

h-I A
(A9.32)

Part V. Appendices

134

In Eq.(A9.32) thc height variable is decomposed in the short wavelength component h > ( r ) and the long wavelength component h<(r). The full Hamiltonian consists of a term which contains only long wavelength components, that with only short wavelength components and that with mixed components as

-

v Jr'dZdyC0S(27r(h< + h>)].

(A9.33)

The potential term with the coefficient V is denoted here H, for short. In the renormalization method, the short wavelength component h>(r) is integrated out and its effect on the long wavelength component h<(r)is systematically studied. The renormalized Hamiltonian 31< contains only the long wavelength variables h<(r),and is defined by

(A9.34) Here Tr>is the trace over the short wavelength component h ( p ) . Actually, the renormalized Hamiltonian 'ti
It is actually a Gaussian average with ( h ( p ) )= 0 and thc hcight corrclation ((h(p)12)= kBT/7a2p2. Average of the term containing the mixed part X, is expanded in the cumulant as

The first order of V can be calculated as

--

kBT

-

knT

J d2r(cos27r(h<+ h'))

(A9.37)

A9 More on Surface Roughening

135

Here the effective potential strength V is V = V exp [ -2n2g(0)] = Vb-*kBT/yaa

(A9.38)

with the height correlation function of the short wavelength defined by

N

-

ksT Jo(Ar) In b, 2aya2

(A9.39)

which decays very fast for r > a / A due to the asymptotic behavior of the Bessel function Jo. The second order term H(2) is calculated as follows:

= ( ~ ) ‘ J d r J d r l g ( r - r l ) j c o s 2 a ( h c + h : ) -cos2a(h‘

-h:)l, (A9.40)

where the notation h = h(r) and hl = h(r1) is used. Since g(r - rl) is almost zero for Ir ~ 1 > 1 a / A , the integrand gives contribution only for T I N T . The first term produces the higher harmonics cos2?r(h< h:) FZ. cos4ah<, and is irrelevant for further discussion. The second term is expanded as

-

JhrlJ0(hlr

+

- r1))cos2a(h<- h:) M const - ~ T ~ A - ~A (Oh‘)’,

(A9.41)

where A is a complicated but a smooth function in the region of interest (1461. Eqs. (A9.40) and (A9.41) shows that the second order perturbation gives the renormalization of the surface tension. The final form of the renormalized Hamiltonian has the same form with the original one as

7-F =

J L dzdy

[$(Oh‘)’

1

- V cos(2sh<)

(A9.42)

where z = b z and 3 = bg. The coupling constant is renormalized from the original ones, ya2 and U = V(s/A)’, to (A9.43)

Part V. Appendices

136

4 3

* 2

I 0

I

0

2

X

Figure A9.3: The renormalization group flow diagram in X = 2ya2/akaT and Y = 4 U / a k ~ Tphase space. The temperature variation of a real system takes place along the dashed line. At low temperatures to the right of the separatrix, the potential strength Y increases by rcnormalization and the surface is smooth. At high temperatures to the left of the scparatrix, Y vanishes by renormalization and the surface is rough.

with X = 2 y a 2 / n k B T . Thcrefore, by coarsening the system with the scale variables X and Y = 4 U / n k ~ Tare renormalized as d X - ___A(2/X)Y2

2 x dY = 2 ( I - + ) Y . dc

dc

E

= In b,

(A9.45a) (A9.45b)

Eqs.(A9.45a,b) are called the renormalization group equations. There is a special fixcd point X " = 1 , Y' = 0, and it will be shown that the critical behavior is controllcd around.this point. Thcrcfore, one can assume that the function A is constant with its argument at X' = 1 as A(2) = A = 0.398. The flow by renormalization in X Y plane-is obtained by dY - 4X-1 (A9.46) dX-A Y ' or after integration the renormalization trajectory is obtained by hyperbola as

4 - ( X - 1)2 - Y2 = -c. (A9.47) A The flow by renormalization is shown in Fig.AS.3. The line Y = 0 for X 5 1 is a line of fixed points. The separatrix C = 0 or &Y/2 = 1 - X corresponds a

A9 More on Surface Roughening

137

critical curve, and terminates at the end of fixed points at X*= 1, Y' = 0. Below the separatrix, the strength of the periodic potential U is renormalized to zero, and the surface is rough in a large scale. Since the physical system has fixed values of surface tension T0a2 and the potential VO,the temperature variation occurs along a line Y / X = 2Uo/70u2 = const. This line crosses the critical line C = 0 at the roughening temperature (A9.48) with t , = 2J;?U0/7ou2.Due to the potential Uo the stiffness 70 is modified to the effective one ye^ = ryo(l + t , / 2 ) , and correspondingly the microscopic coupling J = 7ou2/2 to the effective one .Jeff = y e ~ a 2 / 2The . roughening temperature is given as k B T R = 4Jer/r. When the temperature is off from TR, the parameter C is represented as (A9.49) with t = (2' - TR)/TR. For T < TR,the potential strength Y initially decreases by renormalization until Y, = &, and then Y increases again. The final increment of Y means the enhancement of the pinning potential U to fix the height to an integer value. The surface is smooth. The correlation length is determined as a length scale until Y reaches to the order unity:

<

m

2dY

M -

IT

(A9.50)

Thus the correlation length [ diverges at the roughening temperature TR as

(A9.51) The step free energy /3 is obtained as (A9.52)

A9.6 Exact solution of surface roughening: Body-centered solid-on-solid (BCS0S) model There is an exact solution on the phase transition of the (100) surface of a bodycentered cubic (bcc) crystal (Fig.Ag.4a). There are eight nearest neighbors for each

138

Part V.Appendices

Figure A9.4: (a) Body-centered cubic (bcc) lattice. (b) (100) face of the bcc lattice with the nearest neighbor bonds shown by dashed lines, and the 2nd nearest neighbor by solid lines. ( c )Six possible configurations of surface heights. (d) The corresponding six vertices [187].

A9 More on Surface Roughening

139

atom in the bcc crystal. On looking down the (100) face nearest neighbor bonds form a square lattice (Fig.Ag.4b). The solid-on-solid model assumes that the surface configuration is described by the height of the column on this square lattice. In a bcc crystal the nearest neighboring heights differ by odd numbers in a unit of half the lattice constant. When the nearest neighbor interaction marked by dashed lines in Fig.Ag.4b is attractive and very strong, the height difference is restricted only to unity. Then for a square plaquette there are 6 possible configurations, as shown in Fig.Ag.4~. When one draws a vertical line through the middle of an edge, and draw an arrow such that, when one follows the arrow direction, the right stays higher than the left. Then there are six vertex configurations as shown in Fig.A9.4d, and the model is equivalent to the six vertex model [129]. Among four edges from a vertex, two arrows flows inwards and the other two arrows go outwards. The model is originally introduced to explain the residual entropy of ice. In ice, each water molecule is surrounded by four neighboring water molecules by the hydrogen bonds. On the bond connecting two oxygen atoms, a hydrogen atom can be located near one of them. For each oxygen atoms, two hydrogen atoms lie close to it and two others stay away from it. The vertex in Fig.AS.4d represents the oxygen, and the arrow on the edge indicates that the hydrogen lies close to the oxygen. Thus the rule that the number of incoming and the outgoing arrows are the same is called the ice rule (1531. We now impose a second neighbor interaction, -J, and -Jy in x and y directions respectively, which are shown by solid lines in Fig.Ag.4b. The interaction energies ~1 &6 for vertex configurations 1 to 6 are then given as

-

~1

= €2 = Jy - Jz,

~3

= €4 = J ,

- Jy, and

ES

= &6 = -J, - Jg.

(A9.53)

For an isotropic case, J, = Jy = ~ / > 2 0, the configuration energies are ~1 = c2 = = €4 = 0 and ~5 = &6 = --E.This is called the F model, and the exact solution is known [129]. The transition temperature is known to be

~3

lCBTR - - (ln2)-' E

M

1.4427... ,

(A9.54)

and the singular part of the surface free energy behaves as

Fsjng M exp(-all

- T/T&1'2),

(A9.55)

with o = 7 r 2 / 4 a . Step energies in [lo] and (111directions are also known as

-El111E

m

+ c(-1)"[1+ n=l E

( n - 1) tanh((n - 1)X) - n tanh(nX)]

140

Part V. Appendices

-

T-kBlk

Figure A9.5: Step energies of BCSOS model in two orientations,[lO] (solid line) and (111 (dotted line). For comparison, step energics of the king model is also shown. Dashed line in [lo] direction and dash-dotted line in (111 direction. Crosses represent the difference of the step energies in two directions [187]. 1 3 + z(-l)"[+ ( n - -)43 tanh((n - -)A) 2 4 m

1 4

- ( n - -) tanh((n

n= 1

(A9.56)

with

x = - In t + 2 In (1 + G)

(A9.57)

and t = 4cxp(-2c/lc~T). At low tcmperaturcs, lc~T << E , thc step energy is anisotropic, El111 > E[lol. At high temperatures, lc~T> E , it becomes isotropic. Near the roughening tcmpcrature wherc t + 1 and X x 2M 2 a t / ? k / T - l , step energies behave as E1lll NN E ~ ~ oxl ~X-~/~exp(-7?/2X) NN c x p ( - a / ~ ~ They ~ ) . approach zcro as T + T R - 0, as is shown in Fig.Ag.5. Thc equilibrium shape of facets and thc crystalline shape arc also obtained [92]. Similar correspondence to the F model is found for thc (100) and (110) faces of face centcred cubic (fcc) crystal [93].

141

A14 Advance Rate of a Circular Step

A14 Advance Rate of a Circular Step Detailed calculation of the advance rate of a circular step is given in the paper by Burton, Cabrera and Frank [43], and here I reproduce the calculation. The adatom density or the diffusion field around a circular nucleus with a radius p depends only on the radial variable r as u ( r ) = c ( r ) -em due to the symmetry, and the diffusion equation is written as d2u

dr2

ldu r dr

u =o -

(A14.1)

xZ

in the stationary approximation. The solution which has no singularity at the origin and at infinity is written as

Here 10and KOare the modified Bessel function of the zeroth order. The v-th order ones, I*, and K*”, satisfy the ordinary differential equation

,d2w dw z -+ zd.9

dz

- ( 2 2 + v”w

= 0,

(A14.3)

and I*”(B) is finite at z + 0 and K J z ) is finite at lz( + 00 and larg(z)l < .; v = 0, they can be expanded around z = 0 as

For

(A 14-4)

with Euler’s constant y = 0.57721 56649.. .. Their asymptotic forms for

2 + 03

axe

(A 14.6) (A14.7)

By differentiation, the following relations hold; 1 I ~ ( z=) I~(z), K;(z) = -KI(.z), I o ( ~ ) K l ( d ) I ~ ( z ) K o (= z ).;

+

(A14.8)

Part V. Appendices

142

Tlic local equilibrium is expected around a circular nucleus;

.(PI

(A14.9)

= C e q ( P ) - cco.

On the other hand, the advancc velocity of the step is derived from the material conservation as

(A1 4.10)

When thc radius p is larger than x s , the approximation 1 ~ 0 ( p / z , ) I o ( p / z x ~x) , / 2 p holds from Eq.(A14.6) and (A14.7), and v ( p ) is obtained as

).

D,a2 1 v(p) = -P

X s P P

For the straight stcp with p

+ 00,

v ( p -+

00)

(A14.11)

thc advancc velocity reduccs to vo as

2D,a2 = -(cm

- ceq)= vo.

(A14.12)

5,

Then the velocity of a circular step is approxirnatcd as w(p) = 210 (1

-

);

,

(A14.13)

whcre pc is the critical radius of the two-dimensional nucleation defined in (6.23) as (A14.14)

A15 Advance Velocity of a Spiral Step

143

A15 Advance Velocity of a Spiral Step There are various ways to represent the spiral curve. One way is to use an arc length s along the spiral and the angle 19 of the normal vector n from the y axis as shown in Fig. A1 5.1a. 8 = 8(s). (A15.1) From Fig.Al5.la it is obvious that the radius of the curvature p is related to the variation of the arc length ds and the associated angle change dtJ as ds = pd8, and the curvature is givcn as do ds' The same spiral can be represented by thc polar coordinate at the spiral center as Cp = Cp(r). 1

(A15.2)

K=-=--

p

( T , Cp)

with its origin

(A15.3)

Figure A15.lb shows that the variation of an arc length ds is written as ds =

d

m=d

r d m ,

(A 15.4)

where the prime means the derivative by r: ' = d/dr. There is still another possibility of representing the spiral by the anglc 1/, between the radial vector r and the tangential vector t of the spiral (Fig.Al5.k) [44]:

11, = ">.

(A15.5)

Figure A15.1: Three ways of representing a curved spiral. (a) By arc length s and the angle 8 of the normal vector n making with 9 axis. (b) The polar coordinate (r,q3). (c) The length T of the radial vector r and the angle b, of the tangential vector t and r. Angles are related as 11, = I9 $.

+

144

Part V. Appendices

Figure A15.2: A spiral steadily rotating with an angular velocity w. 6 is the local velocity in the radial direction, and v is velocity normal to the spiral. As is clear from Fig.A15.lc, the three angles are related as $=$+9.

(A15.6)

Also from Fig.Al5.1~it is evident that the relation

r#' = - tan $

(A15.7)

holds. Then the curvature K is written as (A15.8) When the spiral is advancing steadily, the spiral looks as if it is rotating steadily. By denoting the radial velocity at a point (T, #) as 6,the point on the curve at a polar angle $ moves to r1 = r ( t ) I d t (A15.9)

+

after a, time dt. Since the movement looks like a steady rotation with an angular velocity w , the angle @ ( T I , t ) on a spiral at time t should rotate to $ ( r , t ) after a time elapse d t : (A15.10) $(i-l,t) wdt = $(r,t). (See Fig.Al5.2). Inserting (A15.9) into (A15.10) and cxpatiding in terms of a small variable dt, radial velocity 6 is written as

+

u=

w -9"

(A15.11)

A15 Advance Velocity of a Spiral Step

145

The normal component of the step velocity should be (A15.12) and this velocity should coincide with the velocity given by the step curvature as (A15.13) From Eq.(A15.7-A15.8) and fA15.12-A15.13), the spiral first order differential equation tan$ d$ = -dr

+--- 1

T

PCCOS*

111 = $ ( T ) should satisfy the

W

T.

(A15.14)

VOPC

Boundary conditions are set at T -t 0 and at r -, 00: At r -+ 00, the spiral looks like a circle and the angle $J + 71/2. At T -t 0, the radius of the curvature decreases, but it should be larger than pc, because if the radius is smaller than pc, the crystal melt back as is evident from Eq(A15.13). Therefore, at r = 0 the curvature is pc. From (A15.8)II, should approach zero as r + 0 in order to obtain a finite K . Then sin$/r N $ / r d$/dr, and at r + 0 the relation l/pc 2d@/dr holds. Therefore, 11 N r/2pc for small T . Since the angle $ satisfies the first order differential equation (A15.16) the solution contains only one integral constant. Thus two boundary conditions cannot be satisfied in generd. The solution exists only for some special values of w. Thus, Eq.(A15.14) is the eigenvdue equation t o determine the eigenvalue w . From the numerical calculation, the eigenvalue is obtained to be

-

-

w

= o.33vo/pc.

(A15.15)

The period T for one turn of the spiral is T = 2./r/w. Since the advance velocity of the step is vo asymptotically at T -, 00, the step separation X = uoT is calculated as (A15.16) The step separation of a steadily growing spiral is 19 times larger than the critical radius pc of the two-dimensional nucleation. From the relation (A15.7), the polar angle Cp for r 4 0 is determined from the relation r# fi: -$ = -r/2pc to be f$5Y

r 2PC

--,

(A15.17)

This is the Archimedes’s spiral turning right. The spiral turning left is written as Cp = r/2pc, This is used to obtain the formula (15.1).

Part V. Appendices

146

A21 Parabolic Coordinate The Cartesian coordinate ( a ,y, 2 ) and the parabolic coordinate (<, 7, 4) are related

as a = JF;lcos 4, Y = &sin

+

4,

1 = $9 - I),

(A21.1)

+

where r = da2 y2 + z 2 = a(r, [). Inverse relation is written as

t = r + z,

7 = T - z,

#I = arctan-.Y X

(A21.2)

The line element is represented in both coordinate as

+

+

( d ~=) ( ~d ~ ) ( ~d ~ ) ( ~d ~=) (h,dV)2 ~

+ (h&)2 + (h&)2

(A21.3)

with

The gradient vector of the field is

vu

au

du,

= --5+-y+-z ay

ax

au, az (A21.5)

(i, 4)

with basis vectors (i?,$ , 2 ) and ij1 in each coordinate system. The basis vector 2 is related to those of parabolic coordinate as

(A21.6) The Laplacian is written as

(A21.7) By denoting the interface as 9 = ~ i ( [ , d J ) ~the unnormalized tangential vectors are

(A21.8) (A21.9) The normal vector is then obtained as

A21 Perabolic Coordinate

147

with N being determined from the normalization condition In1 = 1. The interface velocity of the dendrite in laboratory frame is written as v = V f ?jh,+j. By using f and n given in Eq.(A21.6) and (A21.10), one obtains the interface normal velocity as

+

[ (-+--

vn=N V

2hf

which is explicitly presented in Eq.(21.3).

2hz

"")+"I, at

(A21.11)

Part V.Appendices

148

A24 Dendritic Growth A24.1 Boundary integral equation for the diffusion-controlled growth In a steady statc approximation, the diffusion equation in the moving frame is written as 2 au Lu 2 v 2 u + -- = 0, (A24.1) 1D az' where z' is a new coordinate z' = z - vt and ID = 2D/v is the diffusion length. Hereafter in this section we drop the prime on z . By using the adjoint operator defined by Efg(r,rl) = V y g - -2 ag (A24.2) In dz, ' the Green's theorem says that the integral over the region 521 enclosed by the periphery rl satisfies the relation:

4

When u(r1 satisfies the diffusion equation (A24.1) and g is a Green's function of the operator L, as (A24 4) i t g ( r , r l ) = -6(r - rl), I

then (A24.3) reduces to the integral equation over the crystal-liquid interface FSL.

J drl,SLg(r,r 1 d)Uz = J drl,sth(r, r1)4r1).

(A24.5)

Here we have used the boundary condition that far from the liquid-crystal interface, the diffusion field and its derivative vanish: ti(r + m) = du/dn[p,oo = 0. The Green's function g is obtained by the Fourier transformation as

with KO being the modified Bessel function explained in Appendix A14. Another integral kernel h is obtained as (A24.7)

The coefficient c enters bccausc the position r in Eq.(A24.5) lies on the boundary rsL. Instead of determining c(r) directly, one can impose a sum rule

A24 Dendntic Growtb

149

(A24.8) The condition is easily derived by inserting the trivial solution u = const of the diffusion equation in Eq.(A24.3) [30, 1651. Equation (A24.5) relates the diffusion field u to its normal derivative du/an at the liquid-crystal interface. This relation is used in the numerical simulation described in Appendix A24.3. If there is a diffusion in the crystal with the diffusion constant Ds, we can obtain another boundary integral equation similar to Eq.(A24.5) with the Green's function gs and an integral kernel hs. In them, the diffusion length is replaced by the diffusion length in the solid Z D , ~ = 2Ds/21,and in hs the normal vector is replaced by ns = -n. Due to the difference of the direction of the normal vector, the sum rule for hs is different from (A24.8) and is written as [168]

The diffusion field is continuous at the boundary; us = u.When the heat is conducted in the crystal as well as in the liquid, the energy conservation boundary condition (18.7) is altered as (A24.10) In a symmetric model where Ds = D , one gets simple relations; 1 ~ =s l ~ gs, = g and h-hs = -b(r-rl). By subtracting Eq.(A24.5) from the corresponding one in the crystal, and by utilizing the above relations and the energy conservation Eq.(A24.10), one gets

=

/aaL [hs(r,rl) - h(r,rl)lu(r1) =

(A24.11)

When the crystal grows steadily in z direction with the velocity vi,the normal velocity is v, = vn, and the integration can be transformed as J dl?l,SI.n, = J dxl. Also by imposing the local equilibrium condition (18.11) or u(r) = A - drc,

(A24.12)

one obtains a simple equation

A

-

=

2 Jm ID

dxlg(r,rl)

(A24.13)

-m

for the symmetric model. This is the relation often used in the theoretical analysis, aa will be explained in Appendix A24.2.

Part V. Appendices

150

A24.2 Microscopic Solvability Theory of Dendritic Growth The solvability theory is very mathematical and its detailed explanation is out of scope of the present notes. I sketch here briefly its main idea and the result [124, 105, 33, 1561. When the crystal is growing steadily, aC/at = 0, and the diffusion constants in solid and melt is the same (symmetric model), the profile z = ((5) of a twodimensional crystal is determined by (A24.13) or

Here KO is the modified Bessel function of the zeroth order, explained in Appendix A14. Without capillarity (d = 0), Ivantsov parabola ( ~ v ( x )= -x2/2p satisfies Eq.(A24.14) at the supercooling A [154]. With the capillarity, the interface [(x) deviates slightly from the Ivantsov parabola, CIV. By using the Ivantsov relation between the supercooling A and the profile (IV and measuring the length in unit of the tip radius p of the Ivantsov parabola, one gets the relation

a A ( x ,C)^: = rZ(p,2 , C) - r2(ex,h>. Here P = p / l n is the Peclet number, the function 1 0 0

r2(p,x,c)= - J lr

dxle-p(c(z)-c(z1))Ko

r2

(A24.15)

is defined by

P J ( ~- x1)2 + ((x)

- c(xl))a> ,

-a3

(A24.16) and

CJ

is the stability parameter defined by

(A24.17) The surface stiffness is assumed to have four-fold symmetry with the anisotropy factor

6 here means the strength of the anisotropy. Since cr is proportional to the small capillary length do, it is also small. But the usual perturbation expansion in o is not applicable, since a couples with the highest derivative of the height or the curvature

(A24.19) One has to use singular perturbation method [16]. The situation is similar to the quantum mechanics [171]: In the Schrodinger equation

(A24.20)

A24 Dendritic Growth

151

the Planck constant h = 27rh acts as asmall parameter, but it is coupled to the highest second derivative of the wave function 9. The approximation to the wave function 9 can be obtained by a singular perturbation, known as WKB approximation [171). Up to the linear order of the deviation of the interface profile from the Ivantsov parabola (A24.21) ~ ( 2-)(+) -= (1 2 ) 3 / 4 2 ( 4 ,

+

Eq.(A24.15) is transformed to the following linear and inhomogeneous integro-differential equation [ 1241

(A24.22) Here P denotes the principal value. For the existence of a nontrivial solution,-the inhomogeneous term in the r.h.s. should be orthogonal to thc zero eigenvector Z ( s ) of the adjoint operator Ct as (A24.23) This equation is an eigenvalue equation to determine the cigenvalue u. B(z) is obtained by the WKB method, and @(a,E ) is calculated to behave approximately as (A24.24) at small u and E. This equation shows that with an isotropic surface tension ( E = 0 ) , 0 can never be 0 for finite u or finite velocity. To realize steady growth, an anisotropy c is necessary in the surface stiffness. For 6 # 0 there are infinitely many but discrete numbers of stationary solutions u ~ u , , c ~ / ~ ( 1 + 2 n(with )~ n=0,1,2,.~.,m).

(A24.25)

From the linear stability analysis around these stationary solutions, the steady state with the minimum u or the maximum velocity v is found to be stable, but the other solutions are unstable. The dendrite tip radius of the curvature p and the growth velocity 'u thus satisfy the scaling relation (A24.26)

152

Part V. Appendices

as well as the Ivantsov rclation (21.13), which reflects the total energy conservation and is influenced littlc by capillarity. The tip radius and the velocity is then determined uniquely for a given supercooling A as (A24.27) and

(A24.28) In the last cxpression in Eq.(A24.27) and (A24.28) Peclct number P ( A ) is approximated by its limit for small supercooling in two dimensions.

A24.3 Numerical Simulation of the Dendritic Profile Sincc the dcndritic pattern is quite different from the flat interface, linear and weakly nonlinear analysis is insuficicnt. To find out tlic strongly nonlinear and far from nonequilibrium pattern, numerical simulation is a very useful method. There arc various different methods to simulated the profile of the crystal [135, 102, 103, 165, 109, 110, 871. The main problem of the simulation is the sharpness of the interface. If one solves the bulk diffusion cquation numerically, a discrete mcsh is necessary. When tlic intcrfacc moves, it will be off from the mcsh points. Sincc the interface is susceptible to tlic fluctuation because of thc Mullins-Sekerka instability, one has to be carcfui in tracing tiic interface motion. Also, from the microscopic solvability analysis, the anisotropy in the system is said to bc csscntial to stabilize t,he dendrite tip. By using the mesh for the diffusion ficld, an artificial anisotropy from the mcsh lattice can be introduced. Thcrc are various simulation mcthods to circumvcnt these obstacles. First we briefly summarize idcas of somc proposed methods. By assuming the stcady growth of the dcndritc, the diffusion cquation can be transformed to the intcgro-differential equation (A24.5) with the help of the Green’s theorem, as cxplaincd in Appendix A24.1. By solving this intcrfacc cquation numerically there o c w r no problems stated abovc, since grid points lic on tlic interface. By intcgrating along one side of a stcadily growing dcndrite from tlic t,ail to the tip, one can search tlic selected velocity such as to producc a round tip which connects smoothly to the othcr side of thc dcndritc symrnctricdly [135, 102, 1031. This is tlie numerical rcalixation of thc solvability criterion of a symnictric dendrite in a stationary code. Our simulation [165] relies on thc intcgro-diffcrcntial equation (.424.5), but it allows the variation of the system velocity so as to select the final profile spontaneously. Thcrc arc othcr mcthuds which don’t usc boundary integral cquation. In a phasc ficld modcl, the interface is dcfincd as thc domain boundary of the phase field and is treated as a diffuse object, in ordcr to gct rid of the difficulty associated to the sharp intcrfacc [log, 110, 111, 1171. This is so far the sole modcl on which a threcdimensional simulation is pcrformcd [log, 11 11. The phasc-field model, however, uses a mesh to solve thc diffusion equation, and thc mcsh-lattice anisotropy is expected

A24 Dendritic Growth

153

to affect the quantitative results. Recently a new algorithm with multiple meshes are proposed to solve the diffusion equation. These meshes are mutually shifted translationally and rotationally [87].With this multiple-mesh method the real dynamical evolution of the crystal interface is simulated in many systems. Here I describe our boundary integral algorithm applied first to study the selection problem of a two-dimensional dendrite profile of a one-sided model, where the diffusion takes place only in the liquid phase [165]. The time evolution of the crystalliquid interface is traced by solving the quasi-stationary distribution of the diffusion field by the boundary element method [30]. First assume that the profile r S L ( t ) and the frame velocity v(t) at time t are given. Then modification of rsL and v to the asymptotic form is performed in the following processes. 1. Calculate the diffusion field u at the boundary from the local equilibrium condition (A24.12), 2. calculate the Green's function g and the integral kernel h by using the frame velocity v,

3. solve the linear equation (A24.5) for the normal gradient q = &/an, since g, h, u and I'sL are known, 4. the normal velocity is calculated by the conservation condition as v,(r) = -Dq(r),

5. then evolve the crystal profile in the normal direction n by r(t+dt) = r(t)+v,ndt to a new configuration rSL(t c dt) at time t dt,

+

6. adjust the moving frame velocity v to the tip velocity of the dendrite vtip by a rather ad hoc relaxation v(t + d t ) = v(t) - ( v ( t )- wtip(t))dt/T,

7. and then iterate back to 1. and repeat the whole procedure until the steady state is realized.

If the steady state is realized, the ad hoc relaxation of the frame velocity by the procedure 6. does not affect the resulting asymptotics. To solve the linear equation (A24.5) numerically by computer, one has to discretize the crystal front into a polygon, whose corner points are denoted by rj with j = 1, . . , N . The diffusion field u and the normal gradient q = &/an at a corner point rj are denoted by uj and qj respectively. Every quantity at a point r on a polygon edge rj between rj and rj+l is interpolated linearly, for example, for u as

4.) = h ( E ) U j + 42(0uj+1

(A24.29) (A24.30)

Part V. Appendices

154

with an interpolation functions PO]

with -1 5 t 5 1. The integration along the edge

rj

is written as

(A24.32) with s j = Irj+l - rj[ being the length of the edge segment equation (A24.5)can be written as a matrix equation

. j ' I

Then the integral

(A424.33)

where

and

21:

+

dth [rj + c ~ , ( t ) s j ri] , #i(t)

J

1

dth [rj - 451(t)sj-i9ri1 (A24.35) with g and h given in Eqs.(A24.6) and (A24.7),respectively. Thc sum rule (A24.8) reduces to the condition N Hij

=

-1

(A24.36) j=l

and the diagonal element Hi, is determined from

H;, = -CH.. '3 - 1.

(A24.37)

j#i

The integration in Eq.(A24.34-A24.35)are performed by the four-point Gaussian quadrature method [I, 301. When the point r, happens to be one of the end points of the linear segment rj where the integration is performed, the Green's function g contains the logarithmic singularity as

(A24.38) with Euler constant 7. Integration then uses the Gaussian integration with a logarithmic singularity 11, 301. Since our main interest lies in the evolution of the dendrite tip, we divide the interface into three parts along z : a tip region, a transition region and a tail region.

A24 Denthitic Growth

155

Evolution of the tip region is treated fully as described. The tail region consists of an Ivanteov parabola appropriate to the steady growth assumption with the frame velocity TI. A transition region connects the tip and tail regions smoothly. Extension of the method to include the kinetic effect or dendrite tilting is straightforward. When the kinetic coefficient K(n) is finite and anisotropic, the local equilibrium assumption does not hold any more. Here one has to use the kinetic condition (18.8) [50]. In the steady.state the diffusion field at the interface u(r1) is decomposed as Vn D du u ( q )= A - d K - -= uo(rl)+ -(A24.39) K(n) K(n) dn with (A24.40) uO(r1) = A - dh;. The integral equation (A24.5) is now modified as (A24.41)

The kinetic effect introduces only a small modification in the integral kernel as g D h / K , which is readily calculated. The growth rate vn of the interface is obtained from the normal gradient au/an as in Eq.(18.7). Therefore, a simple modification of the integral kernel Gij can include the kinetic effect (50, 1691. In the directional solidification, the surface takes various periodic structures. In order to simulate an interface evolution with a periodic modulation with a periodicity A, one imposes a periodic boundary condition ((s + A) = <(s) in x direction. For numerical simulation, one period of interface is decomposed into N mesh points. Periodic images of the interface are taken into account in the kernel as (A24.42) j=1

with

j=1

m=-m

c M

00

. G..'3 - C G;~+,,,N, and

Hij

=

Hi,j+m~

(A24.43)

m=-m

In practice, .the influence of the m-th image far from the system in consideration are negligible. We usually take the effect of images within the range of 531, [167, 501. For a dendrite in a channel with impermeable walls, the normal gradient of the diffusion field at the wall vanishes from the symmetry: &/an = 0. This means the existence of mirror images at both walls: ((-x) = ((2X - x) = ((s) for a system within a channel of width A. Then the system has a periodicity 2X and one can appropriately modify Eq.(A24.42) in this case. In the unidirectional solidification, the temperature gradient is applied on the system. If orientations of the temperature gradient and of the crystalline axis are different, the tilted structure appears [2, 661. In this case the tip of the dendrite

Part V. Appendices

156

shifts transversally, and the locus of the tip is tilted from the temperature gradient by an angle 4.The system is invariant in a moving frame with a transversal velocity v, = vtan4 and a vertical pulling velocity vz = v. The diffusion equation in quasistationary approximation reads as (A24.44)

The Green's function g and an integral kernel h are now modified to

[

g(r,rl) = 1 exp - (5 - 51) - (z - Zl)tan4 27r I,,

(x - zd2 + (5 - C1)2 (A24.45)

and

Further discretization and numerical simulation can be performed similarly as before [148].

A26 Eutectic Growth Theory by Jackson and Hunt

157

A26 Eutectic Growth Theory by Jackson and Hunt The surface supercooling AT at the eutectic crystal front is derived by Jackson and Hunt [Sl).They considered both the lamellar and rod structure of an eutectic system, but here only the lamellar structure in two dimensions is summarized. The eutectic crystal is growing from the liquid with a concentration c,. Concentrations of the crystal QI and p phases are cg and the periodicity of lamellae is A, and the ratio of each phases are set 17 and 1 - q respectively. The material conservation requires the relation

4,

4Q- 17))

c, =

(A26.1)

(4

and thus 17 = - c,)/Ac and 1 - 9 = (c, - cg)/Ac with the miscibility gap Ac=r$-c$ (>O). The dimensionless concentration field u defined by (A26.2)

satisfies the diffusion equation in the liquid (A26.3)

The last equality holds in the quasi-stationary approximation. The diffusion length in the liquid is defined by means of the chemical diffusion constant D, as 20, ID = -. (A26.4) U

In solid the material diffusion is assumed negligible (one-sided model). The thermal diffusivity is so high in all the phases that the temperature gradient is constant as Eq.(26.1): GT = d T / d z = const or T ( z )= TE GTZ. The crystal-liquid interface is denoted as z = ((x,t). Then the material conservation at the boundary yields

+

for 0 < .z < qX in

Q

phase (A26.5)

- [u(z,4 ) - u:]

for VX < z < X in /3 phase,

-

where uTp = (c;?@ c ~ ) / A c . The local equilibrium assumption at the interface is written 89 &Z'(z)

TE- T(C)= -GT(

=I

+

z?z~Acu(x,C) YL ~ ~ E ~ E for o < x < VX La (A26.6)

-mpAcu(x,<)

+* K .

L5

for q X < x < A.

Part V. Appendices

158

Here m,,p = \dTt,/dct7filis the absolute slope of the liquidus lines with QI and crystal phases, K. is the curvatures, y ~and , ~yr,fi are the surface tensions between the liquid and a or p phase, L , and Lo are the latent heat of a and ,B phases respectively. From Eq.(A26.6) it is obvious that to obtain the supercooling AT at the average interface position (0, we have to know the average concentration field (u) and the average curvature (K). The field u is expected to be modified periodically with the same periodicity X of the interface modulation. Thus u can lie expanded in a Fourier series as W

~ ( 2 , x= )

'u,

+

Bnc-*"("-(C))ciqnz,

(A26.7)

n=-w

where qn = 2rn/X. Since u has to satisfy the diffusion equation in the quasi-stationary approximation (A26.3), the decay rate A, in z direction is found to be (A26.8)

The continuity equation (A2G.5) is written at the average position 2 A,Bneiqnx= - [Au(x) 1D

where

Au(x)=

{ 'iV

+

B,eiqnr],

for 0 < x iVX for QX < x < A.

(0as (A26.9)

(A26.10)

Since the Fourier coefficicnt of AIL(.) is calculated as (A26.11) the coefficient B,, for ?t # 0 is determined as (A26.12) Assuming the slow growth such that X << I D , the average conccntration in front of the a or ,B phase is calculatcd as

where (A26.14)

A26 Eutectic Growth Theory by Jackson and Hunt

159

From Eq.(A26.13) one obtains (A26.15)

or (A26.16) The growth velocity Eq.( A26.16) corresponds to the phenomenological expressions (26.3) and (26.5) with Qa in Eq.(26.5) at ,C = CF,being given as (A26.17) In order to obtain the average curvature ( K ) , we use the relation of the force balance at the triple point, where three phase, liquid, Q and p, meet. (See Fig.26.4). The surface tensions satisfy the relation (26.6a,26.6b) or:

nasin 8,

t YLD sin 60 = ~ " y a g TI," cos 6" - yLp cos 60 = 0,

(A26.18)

where the angles 8, and 8, are defined in Fig.26.4. These contact anglcs give the average curvature as

(A26.19) Here we used the relation (A15.2) of the curvature K , the anglc 0 of the normal vector and the arclength s: K = dO/ds. These curvatures are the same with those obtained in Eq.(26.7) by assuming that the interface is a part of circlc. F'rom Eq.(A26.6) the average supercooling at the interface, z = is expected to be constant for a and /3 phases as (AT") = (ATP) = AT. Then the rcmaining unknown constant BOis determined as

(c)

The interface supercooling AT is obtained as

Part V. Appendices

160

(A26.21)

Here (A26.22)

is the minimum value of the interface undercooling, and (A26.23)

is the corresponding pcriodicity of thc lamellar structurc. The capillary lengths d, and dp are defined as

d, =

YI,uTE

m, L, Ac

~

and

dp =

rr.pTE ~

Lp Ac'

These equations (A26.21-A26.24) are more general than Eq.(26.10).

(A26.24)

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Index dendrite CD 99 seaweed CS 99 Concentration distribution around polygon 121 in solution 103,158 of adsorbed atom 46-51,61 Conservation law

a-parameter 26,27 Absolute stability 107 Activation energy + Energy barrier Adjoint operator 148,151 Adsorbed atom 20,30,46-51,58-G2 Aggregation 82-85 Alloy growth 101-120 Amplification rate G1,78-81,95,10G-110 Amplitude modulation 109 Angle of normal vector

48,62,68,73,76,104,149,153,157

Convective stability 94 Convex envelope 99 Correlation length 29,132,137 Critical curve 137 nucleus size 52 point 16 radius 14,51,72,145 velocity 106 wavcnumber 107 Cumulant 134 Curvature 14,61,68,143,150,158 in spherical coordinate 80 radius 15,117,143 cusp siiigularity 41 structure 111

13-17,33,86,90,113,143,144

Anisotropy 14-1736-90,93,113,150-152 factor 150 strength of 14,86,92,151 Arc length 14,33,143 Archemedes’s spiral 51,145 Aspect ratio 75 Attachment kinetics 58 Bacteria colonies 82 Berg effect 120-122 Bifurcation subcritical 109 supercritical 109 Body-centered solid-on-solid (BCSOS) model 137 Boltzmann weight 6,9,24,38,53,100 Boundary element irrctliod 88,111,119,123,1~3-156 Boundary integral 148,149,155 Brownian iiiotion 39

Deiitlrite 66,111 asyiirmetric 98 free 86 in a clianiiel 97 irregular 82,99 regular 86,100 symmetric 88,97 three-dimensional 90-93,97 tip 86 two-dimensional 86-90,93-97,153 Dense l~raiichedmorphology DBM 99 Deposition 8,4G,59 flux 8,46,48 critical rate 60,64

Capillary effect 86.100 length 68,86,105,113,117,15O,16O Cellular structure 111 Chemical potential G,8,9,22 difference 6-9,21,24,52,55 Coexistence curve 4,22 Compact 83,84 174

Index

Desolvation 9 Desorption 8 Detailed balance 24,53 Diffusion chemical 46,GG,82,99,101, 116,121 constant 6,47,104,149,150 equation 47,66,68,104,148,156,157 in parabolic coordinate 73,7G in spherical coordinate 71 dimensionless field 68,104,157 length 69,77,84,93,105,117,148,157 in solid 149 local 95 edge 20 surface 20,30,46,59,82 Diffusion-limited aggregation, DLA GG,83 Diffusional instability 60,62,77,105,121 Dispersion relation G2,79,82,106 Doublon 98,100 Eckhaus boundary 110 instability 108 Eigenvalue 40,110,145,151 equation 40,110,145,151 Eigenvector 40,151 Einstein-Stokes relation G Electrochemical deposition 82,85 Electrostatic potent,ial 121 Elliptic paraboloid 75 Energy interatomic 3,21,24,31,124 interface 10 Energy barrier for desolvation 9 for diffusion G for surface diffusion 46 Enthalpy 4 Entropy 4 reduction 38,39 Equiconcentration 121 Equilibrium shape 10-17,41,100 Equipartition law 34

175

Error function 70,75 Euler-Lagrange equation 11 Eut ect ic growth 114,157 Jacksoii- Hunt theory 119,157 point 114 Evaporation 46,59 Exact solution 137 . Exponential integral function 75 F niodel 30,139 Facet 22,42,120,121 arm 123 size 42 Facet ing transit ion 20,4 1,42 Fin 92,97 Fixed point 137 Force balance 116,159 Four-fold symmetry 150 Fractal 82,84,95,101 dendrite FD 101 diinension 83 forest 84 seaweed FS 101 Free energy barrier for nucleation 53 density 11 inequality 130 of step 13,17,28,30-34,38-42, 50,133,137,139 of surface 11,22,38,41,43, 86-93,111,116 Gas density 84 Gaussian model continuous 128 discrete (DG) 124 Gihbs free energy 4 Giblx+Thoinson effect 18,50,59,77, 8 8 , 1 0 4 17 ~ Green’s function 148,153,156 latticc 129 Green’s theorem 148

176

Growth law from melt 7 from vapor 8 from solution 9 linear 7,8,9,52,5G parabolic 52 rate 72,75 by spiral 51 by two-dimensional nucleation 55 of dendrite 152 of eutectics 119 shape 17,18 Gruber-Mullins model 39

lndex

Ivaiitsov parabola 72-76 $8,150 relation 74,86,88,92,93 Kinetic coefficient 7-9,18,25,59,67,68,120,155 resistivity 72 Kinetics-limited growth 46 I
Lamellar periodicity 118 s t ruc t urc 114,157 Landau coefficient 108 Hamiltonian 24 Landau-Ginzhurg expansion 108 effective 131 Laplace equation 83,121 of contiuuous model 133 Laplacian in parabolic coordinate 146 of Ising model lG,22 Latent heat 4,7,20,66,104,117,158 of SOS model 24,31,124 Lattice gas model lG,21,99 of Gaussian model 124 Legendre transformation 12 of modified Gaussian model 129 Lifetiiiie of adatoin 46 Heat conduction 7,GG,102 Linear flux 77 stability analysis G1,79,105,151 Height 23,29,124,133 iiiterpolatioii 153 correlation 135 response 17 difference correlation 29,33,124,129,133 Liquidus line 102,117,158 contiiiuous 128,130 silope of 103,117 integer 23,124 local equi1if)rium 47,59,68,72,74,76,77, long wavelength component 134 104,I 15,142,149,153,157 short wavelength component 134 Hele-Shaw cell 90,95,101,114 Master equation 24 Hertz-I
177

Index

Molecular beam epitaxy 7 Monte Garlo dynamical25 simulation of DLA in a finite-density gas 85 of growth shape 99 of roughening 22-26,30,56,124128 of steps 64 step 25 Morphology phase diagram 98,100 transition 99,100 Most unstable mode 80,95 Moving frame 69,148,153,156 Mullins-Sekerka instability 77,88,93 Multinucleation growth 55 Multiple-mesh method 153 Neutral curve 107,110 Noise amplification 97 effect on dendrite 93 origin of sidebranches 94 Normal gradient 122,153 vector 11,12,38,61,143,146 velocity 61,67,95,147,153 Nucleat ion rate 54,55 two-dimensional 30,52,123 One-sided model 59,67,88,153,157 Orientation dependence --t Anisotropy Oscillation instability 120 Parabolic coordinate 73,76,146 crystal 72,74,93 Parity breaking 119 Part it ion coefficieiit 103

function 32,36,39,128 Pattern formation 66 Peclet number 75,150 Periodic boundary condition 155 structure 102,155 Periodicity 114,160 of sidehranch 95 Perturbation expansion first order 134 second order 135 Phase diagram 102,114 morphological 98,100 field model 152 modulation 110 transition 3,26,132 first-order 5 liquid-crystal 4 of surface 20 Polyhedral crystal 1,120 Quasi stationary approximation 89,106, 148,157 Radial velocity 144 Random walk 46 Rate equation 53 Reflection electron microscope 31 Relaxation dynamics 25 Renormalization group equation 13G theory 29,133 trajectory 13G Roughening temperature 27,30,33,37,41,56,133, 137,139 transit ion 20,29,41,133 exact solution 29,137 mean field approximation 26,29 inultilayer model 28 renormalization group theory

178

29,133 single layer model 2G variational method 129 Roughness 7,21 Scaling form 69 relation 88,119,151 Scanning tunneling microscope 22,31 Schrijdinger equation 150 Schwoebet effect 59 Screw dislocation 30,46 Segregation coefficient 103 Self-similarity 83 Separatrix 137 Sidebranch 93-97 correlation of 94 Simulation 120 by boundary element inethod -+ Boundary element method by Monte Carlo method -+ Monte Carlo Single nucleation growth 58 Singular face 38,120 Singular perturbation method 88,150 Six vertex model 30,139 Slow dynamics 63 Snow 2,120,123 Solid-on-solid (SOS) niodel23,25,29,124 Solution growth 9,67,101 Solvability condition for a two-dimensional system 8G,88,93, 93,98,150,153 for a three-diiiiensional dendrite 90,92 of doublon 98 Spatio-temporal chaos 64 Spherical crystal 71,80 harmonics 80 Spiral growth 31,51,143 step 143 Stability

hadex

length 80,88 of flat interface 77,105 of spherical crystal 80 of straight step 59 parameter 88,150 Stationary approximation 50,61,71,141 condition 11,121 Steady state + Quasi-stationary approxh a t ion Step 20,28,30,38,41,46,59 circular 49,50,141 collision 39 confinement 39 density 123 free energy -+ Free enrgy of step fluctuation 30,33 isolated 47,48 separation 51,145 stability 61 stiffness 37 train 48,49 unstabie GI velocity circular 50 isolated 48 train 49,51 width 34 Step-step interaction 38-41 Stiffncss effective G3 kinetic 18 parallcl component 43 pcrpciidicular component 43 of stcl, 15,16,33-35,60 of surface 13,14,42,G8,86,113,133 Stochastic process 22 Succinonitrilc dendrite 90 Suiii rule 148,149 Siipxooliiig + Undercooling SIIr face diffusion length 47,50,59 tliffusivity 47

Index

kinetics 46,66,113 supercooling 157 tension 13,72,135,158 isotropic 151 Symmetric model 80,88,149,150

179

Weak nonlinearity 63 Wilson-Fkenkel formula 7,67,68 Wulff plot 13 growth shape 18 Xenon dendrite 93

Tangential vector 143,146 Temperature gradient 102,115,157 Terrace 20,38,59 Thermal conductivity (57,102,149 diffusivity 68 Auctuation 22,28 gradient 102,113 length 105 Thermodynamics first law 4 second law 4,s Tilted growth 113 structure 113,155 lamellar 119 Tilting field 35 Tip radius 75,86,88,150 splitting 64,89,95,100 stability 90,93 Transfer matrix 39 Transition probability 23 Transversal velocity 156 Triple point 114,116,159 Undercooling 68,75,86,117 Unidirectional solidification 101,114,155 Upper critical velocity 107 Vapor growth 7 Variation method 129 Velocity selection 86 Vicinal face 38,41,42,48,122 Viscosity 6 Viscous finger 90 WKB approximation 151

Zeldovich factor 55